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THE   MODERN   MATHEMATICAL   SERIES 

LUCIEN   AUGUSTUS   WAIT  .  .  .  General  Editok 

(senior  PKOFE8SOB  OF  MATUEMATIC3  IN  COKNELL  UNIVEESITY) 


The  Modern  Mathematical  Series, 
lucien  augustus  wait, 

{Senior  Professor  of  Mathematics  tn  Cornell  University,) 
GENERAL   EDITOR. 


This  series  includes  the  following  works : 

ANALYTIC  GEOMETRY.    By  J.  H.  Tanner  and  Joseph  Allen. 
DIFFERENTIAL  CALCULUS.    By  James  McMahon  and  Virgil  Snyder. 
INTEGRAL  CALCULUS.    By  D.  A.  Murray. 

DIFFERENTIAL  AND  INTEGRAL  CALCULUS.    By  Virgil  Snyber  and  J.  I. 

Hutchinson. 
ELEMENTARY  ALGEBRA.    By  J.  H.  Tanner. 
ELEMENTARY  GEOMETRY.    By  James  McMahon. 


The  Analytic  Geometry,  Differential  Calculus,  and  Integral  Calculus  (pub- 
lished in  September  of  1898)  were  written  primarily  to  meet  the  needs  of  college 
students  pursuing  courses  in  Engineering  and  Architecture ;  accordingly,  prac- 
tical problems,  in  illustration  of  general  principles  under  discussion,  play  an 
important  part  in  each  book. 

These  three  books,  treating  their  subjects  in  a  way  that  is  simple  and  practi- 
cal, yet  thoroughly  rigorous,  and  attractive  to  both  teaclier  and  student,  received 
such  general  and  hearty  approval  of  teachers,  and  have  been  so  widely  adopted 
in  the  best  colleges  and  universities  of  the  country,  that  other  books,  written  on 
the  same  general  plan,  are  being  added  to  the  series. 

The  Differential  and  Integral  Calculus  in  one  volume  was  written  especially 
for  those  institutions  where  the  time  given  to  these  subjects  is  not  su£ficient  to 
use  advantageously  the  two  separate  books. 

The  more  elementary  books  of  this  series  are  designed  to  implant  the  spirit  of 
the  other  books  into  the  secondary  schools.  This  will  make  the  work,  from  the 
schools  up  through  the  university,  continuous  and  harmonious,  and  free  from 
the  abrupt  transition  which  the  student  so  often  experiences  in  changing  from 
his  preparatory  to  his  college  mathematics. 


ELEMENTARY  ALGEBRA 


I 


BY 


J.    H.    TANNER,   Ph.D. 

ASSISTANT    PROFESSOR   OF   MATHEMATICS   IN   CORNELL   UNIVERSITY 


NEW  YORK-:.  CINCINNATI.:.  CHICAGO 

AMERICAN    BOOK    COMPANY 


T-3 


COPTEIGHT,    1904,    BY 

J.   H.   TANNEE. 

Entered  at  Stationers'  Hall,  London, 
tanner's  elem.  alg. 


(^A/'i,^       P.. 


PREFACE 

In  writing  this  book  one  of  the  chief  aims  of  the  author  has 
been  to  make  the  transition  from  arithmetic  to  algebra  as  easy 
and  natural  as  possible,  and  at  the  same  time  to  arouse  and 
sustain  the  student's  interest  in  the  new  field  of  work. 

Accordingly  the  first  few  pages  are  devoted  to  a  restatement 
and  slight  extension  of  the  meaning  of  the  ordinary  arithmetical 
operations.  Then  the  literal  notation  is  introduced,  and  the 
innovation  immediately  justified  by  showing  that,  among  other 
advantages,  it  enables  the  student  to  solve  with  ease  a  class  of 
problems  which,  by  unaided  arithmetical  analysis,  had  previously 
been  very  difficult  for  him. 

In  Chapter  II  negative  numbers  are  introduced,  but  only  after 
it  has  been  shown,  by  concrete  examples,  that  these  numbers  are 
essential  to  man's  needs,  and  that  they  arise  naturally  from 
positive  numbers.  Moreover,  to  make  this  extension  of  the 
number  system  seem  less  startling,  it  is  pointed  out  that  an 
altogether  similar  extension  has  already  been  made  in  arithmetic 
by  the  introduction  of  fractions. 

And  so  on  throughout  the  book,  wherever  an  essentially  new 
step  is  to  be  taken,  its  naturalness  and  advantages  are  presented 
with  it,  and  it  is  thereafter  freely  employed  until  it  becomes  a 
useful  tool  in  the  student's  hands. 

Moreover,  in  order  to  avoid  every  unnecessary  discouragement 
to  the  student,  the  proofs  of  the  various  principles  involved  in 
his  work  are  deferred,  not  only  until  after  he  has  correctly 
apprehended  and  freely  employed  those  principles,  but  also  until 
after  he  has  been  convinced  of  fjfce  necessity  of  a  proof;  compare 
§§  49,  62  (note),  95,  146  (footnote),  176,  etc. 

Another  important  object  of  this  book  is  to  teach  the  student 
to  think  clearly.  "There  is  considerable  danger  of  the  true 
educational  value  of  arithmetic  and  algebra  being  seriously  im- 
paired by  reason  of  a  tendency  to  sacrifice  clear  understanding 
to  mere  mechanical  skill."  *     The  mere  manipulation  of  algebraic 

*  From  the  report  of  a  Committee  of  the  London  Mathematical  Society  ap- 
pointed to  consider  the  subject  of  the  teaching  of  elementary  mathematics. 


137919 


VI  PREFACE 

symbols,  however  cleverly  performed,  is  of  no  advantage  what- 
ever in  after  life  to  the  vast  majority  of  those  who  study  algebra 
in  the  schools ;  but  the  training  in  correct  reasoning  and  in  an 
appreciation  of  the  validity  of  conclusions  that  may  be  drawn 
from  given  data,  which  algebra  wheh  rightly  taught  affords,  is 
of  vast  importance  to  every  one. 

Accordingly,  although  the  early  part  of  each  new  topic  has 
been  presented  as  concretely  and  simply  as  possible,  and  although 
the  student  has  been  led,  often  without  conclusive  proofs,  to 
infer  correctly  the  principles  involved  and  to  perform  the  various 
operations  freely,  his  attention  has  always  been  called  to  the 
fact  that  results  obtained  in  this  way  must  be  regarded  as  tenta- 
tive until  after  the  proofs  have  been  given;  and  the  discussion 
of  no  topic  has  been  finally  closed  without  a  rigorous  demonstra- 
tion of  all  the  principles  involved  therein. 

New  topics  have  always  been  brought  in  where  they  were 
needed,  and  this  has  made  it  necessary  in  some  cases  to  defer 
the  final  proofs  considerably  (cf.  Chapters  VI,  XVIII,  and  the 
Appendices) ;  this  arrangement  has  the  further  advantage,  how- 
ever, of  making  it  possible,  if  the  teacher  prefers,  to  omit  the 
harder  proofs  altogether  on  a  first  reading,  without  breaking  the 
continuity  of  the  subject. 

While  this  book  is  designed  to  meet  the  most  exacting  entrance 
examination  requirements  in  Elementary  Algebra  of  any  college 
or  university  in  this  country,  and  especially  the  excellent  revised 
requirements  of  the  College  Entrance  Board,  yet  the  arrangement 
of  the  book  will  be  found  to  be  peculiarly  suited  to  a  briefer 
course  where  that  should  be  desired. 

The  author  takes  pleasure  in  acknowledging  his  indebtedness 
to  his  colleagues  in  Cornell  University  for  valuable  suggestions, 
especially  to  Professors  Wait  and  McMahon,  who  have  read  both 
the  manuscript  and  the  proof-sheets;  to  Miss  Lelia  J.  Harvie, 
formerly  of  the  Virginia  State  Normal  School,  who  assisted  in 
preparing  and  grading  the  exercises  in  a  large  part  of  the  book ; 
to  Dr.  William  J.  Milne  of  the  State  Normal  College,  Albany, 
N.Y.,  for  his  kind  permission  to  make  free  use  of  the  exercises 
in  his  books ;  to  Professor  H.  W.  Kuhn  of  the  Ohio  State  Univer- 
sity, and  to  several  colleagues  in  the  secondary  schools,  whose 
advice  has  been  helpful. 


CONTENTS 

[See  also  Index  at  the  end  of  the  book.] 

ARTICLES                                   I.     INTRODUCTION  page 

1-  4.   Number.     Arithmetical  processes 1 

5-  8.    Literal  notation  ;  operations  witli  literal  numbers      ...  5 

9-10.  Advantages  of  literal  notation.     Recapitulation          ...  16 

II.     POSITIVE  AND  NEGATIVE  NUMBERS 

11-15.   General  remarks  ;  negative  numbers  defined  and  interpreted     .  18 
16-20.    Operations  with  negative  numbers      .         .         .         ,         .        .23 

21-22.  Algebraic  expressions.     Recapitulation 30 

III.     EQUATIONS  AND  PROBLEMS 

23-25.   Definitions.     Directions  for  solving  equations     ....  32 

26.   Problems  leading  to  equations 36 

IV.     ADDITION  AND   SUBTRACTION  —  PARENTHESES 

27-28.   Definitions :    monomials,   polynomials,   positive    and   negative 

terms,  etc 42 

29-30.   Addition  of  monomials  and  of  polynomials         ....  44 

31-32.    Subtraction  of  monomials  and  of  polynomials     ....  46 

33-35.   Parentheses  ;  removing  and  inserting  parentheses      ...  49 

V.     MULTIPLICATION  AND   DIVISION 

36-38.   Multiplication.     Law  of  exponents.     Product  of  monomials      .  52 

39-40.   Product  of  polynomials 55 

41-42.    Multiplication  with  arranged  polynomials  ;  detached  coefficients  58 

43-44.    Division.     Law  of  exponents.     Negative  and  zero  exponents     .  62 

45-47.   Division  with  monomials  and  with  polynomials          ...  64 

48.   Remainder  theorem 71 

Review  questions  on  Chapters  I-V 72 

VL     COMBINATORY  PROPERTIES   OF  NUMBERS 

49-51.    Commutative  and  associative  laws  of  addition    .         .         .         .  74 

52-53.    Commutative  and  associative  laws  of  multiplication  ...  77 

54.  Fundamental  principles  in  operations  with  fractions  ...  80 

55.  Zero  ;  operations  involving  zero 84 

VII.     TYPE   FORMS  IN  MULTIPLICATION  —  FACTORING 

56-61.   Various  type  forms  of  products 87 

62.   Binomial  theorem 92 

vii 


Ylll 


CONTENTS 


63-  66. 

67. 

68. 
69-  71. 

72. 

VIII. 

73-  75. 

76-  78. 

79. 
80-  82. 


Various  type  forms  in  factoring 94 

Factc^ring  by  means  of  the  remainder  theorem        .         .         .  100 

Binomial  factors  of  x"  ±  a**      . 102 

Other  devices  for  factoring 105 

Solving  equations  by  factoring 109 

HIGHEST  COMMON  FACTORS  — LOWEST  COMMON 
MULTIPLES 

H.  C.  F.  by  means  of  factoring 112 

H.  C.  F.   of  expressions  which  can  not  be  readily  factored ; 

demonstration  of  principles  involved         .         .         .         .114 

An  expression  can  be  factored  into  primes  in  but  one  way     .  120 

L.  C.  M.  of  two  or  more  algebraic  expressions         .        .        .  122 


IX.     ALGEBRAIC   FRACTIONS 

83-  88.   Transformation  of  fractions 126 

89-  93.    Operations  with  fractions         .......  131 

Review  questions  on  Chapters  VI-IX 139 


94-  95. 
96-  97. 


99. 
100. 


101-103. 
104-107. 
108-109. 
110-111. 
112-113. 
114-116. 


X.     SIMPLE   EQUATIONS 
Introductory  remarks.     Equivalent  equations 


^ 


117-118. 
119. 


120-121. 
122-124. 

125. 

126. 
127-128. 

129. 


141 


Literal  equations.     A  simple  equation  has  one  and  only  one 

root 145 

Fractional  equations 147 

Demonstration  of  principles  involved  in  §  98.     Problems        .  149 

General  problems.     Interpretation  of  results  ....  157 


XL     SIMULTANEOUS   SIMPLE   EQUATIONS 

Indeterminate  equations  . 

Simultaneous  equations.     Elimination    .... 

Principles  involved  in  elimination 

Fractional  equations  ;  literal  equations   .... 
Systems  of  equations  containing  three  or  more  unknowns 
Graphic  representation  and  solution  of  equations    . 

XII.     INEQUALITIES 
Definitions  and  general  principles 


Conditional  and  unconditional  inequalities 
Review  questions  on  Chapters  X-XII 

XIII.     INVOLUTION   AND   EVOLUTION 

Involution.     Exponent  laws    . 
Evolution.     Roots  extracted  by  inspection 
Square  roots  of  polynomials     . 
Sciuare  roots  of  arithmetical  numbers 
Cube  roots  of  polynomials  and  of  numbers 
Higher  roots  of  polynomials  and  of  numbers 


162 
165 
170 
174 
183 
189 


193 

197 
200 


201 
205 
209 
213 
216 
221 


CONTENTS 


IX 


XIV.     IRRATIONAL  AND  IMAGINARY  NUMBERS  — 
ARTICLES  FRACTIONAL   EXPONENTS 

130-132.   Irrational  numbers  ;  preliminary  remarks  and  definitions 
133-135.    Product  and  quotient  of  radicals  of  the  same  order 

136-139.    Transformation  of  radicals 

140-144.    Operations  with  radicals 

145.    Important  property  of  quadratic  surds 

146-150.    Imaginary  numbers,  and  operations  with  them 

151.  Important  property  of  complex  numbers         .         .         .         . 

152.  Complex  factors.     Solving  equations  by  factoring  . 

153-154.    Fractional  exponents 

155-159.   Demonstration  of  exponent  laws  with  fractional  exponents ; 

summary  of  these  laws 

160.  Operations  involving  fractional  exponents       .         .         .         . 

161.  Rationalizing  factors  of  binomial  surds 

XV.     QUADRATIC   EQUATIONS 

162-163.  Introductory  remarks  and  definitions 

164-166.  Solution    of    quadratic    equations,  —  by    "completing    the 

square,"  by  factoring,  and  by  formula     . 

167-168.  Character  of  the  roots  ;  their  sum  and  product 

169-170.  Fractional  and  irrational  quadratic  equations 

171.  Problems  which  lead  to  quadratic  equations    . 

172.  Equations  solved  like  quadratics 

173.  Maxima  and  minima  values     .... 
174-179.  Quadratic  equations  in  two  unknowns ;  various  devices  for 

solving 

180.    Systems  containing  three  or  more  unknowns  . 
181-182.    Square  roots  of  quadratic  surds  and  of  complex  numbers 
183-185.    Graphic  representation  and  solution  of  quadratic  equations 


XVL     RATIO,   PROPORTION,   AND  VARIATION 

186-187.    Ratio.     Incommensurable  numbers         .... 

188-189.   Proportion;  definitions  and  principles    .         .         .         . 

190.    Variation.     Constants  and  variables        .... 


PAGE 

223 
228 
233 
237 
243 
244 
250 
251 
252 

255 
261 
264 


268 
277 
282 
286 
291 
294 

297 
308 
310 
314 


318 
320 
327 


191-194. 
195-198. 

199. 

200. 


XVIL     SERIES.     THE  PROGRESSIONS 

Series.     Arithmetical  progression    .         .         .         , 
Geometric  progression      ...... 

Aritlimetico-geometric  series 

Harmonic  progression 


331 
336 
342 
342 


XVin.    MATHEMATICAL  INDUCTION— BINOMIAL  THEOREM 

201.    Proof  by  induction 344 

202-204.    The  binomial  theorem 346 

205.    The  square  of  a  polynomial 350 

Appendix  A.     Irrational  Numbers 351 

Appendix  B.     Complex  Numbers 365 


NOTICE 

It  is  not  expected  that  pupils  will  be  asked  to  solve  all  of  the 
very  large  number  of  exercises  and  problems,  but  rather  that  the 
teacher  will  make  such  selections  as  will  best  suit  the  needs  of 
his  or  her  classes. 

If  the  teacher  desires  a  briefer  course  than  that  provided  in 
the  book,  or  prefers  to  omit  the  proofs  on  a  first  reading,  the 
following  articles,  together  with  their  attached  exercises,  may- 
be omitted  without  breaking  the  continuity  of  the  work : 


Articles    50-54 

Pages    74-83 

Omit  exercises  1-14,  pp.  84- 

u 

77-79 

116-122 

(I 

95 

143-144 

Take  exercises  5-15,  p.  145 

a 

99 

149-150 

"           "         3-6,    p.  151 

t( 

103 

163-164 

(C 

108-109 

170-172 

Take  exercises  on  p.  173 

(( 

114-116 

189-192 

(( 

127-129 

216-222 

Notes 

1-2 

285 

Omit  exercises  17-22,  p.  28 

Articles    173 

294-297 

(I 

176 

298-299 

(C 

183-185 

314-317. 

The  teacher  will  also  find  it  easy  to  abbreviate  somewhat  the 
work  of  Chapters  XIV  and  XV. 

If  the  above  omissions  are  made,  it  will  be  necessary  to  pass 
over  a  few  isolated  exercises  and  notes  such  as  Ex.  3,  p.  184,  and 
note  1,  p.  301,  and  also  to  change  slightly  the  headings  to  some 
sets  of  exercises  such  as  those  on  p.  145. 


/  OrTHf 


ELEMENTARY  ALGEBRA 

CHAPTER   I 
INTRODUCTION 

1.  Algebra  may  be  regarded  as,  in  a  certain  sense,  a  continuation 
and  extension  of  arithmetic ;  it  may  be  best,  therefore,  to  recall 
briefly  the  subject  matter  and  some  of  the  processes  of  arithmetic 
before  taking  up  the  study  of  algebra. 

It  will  presently  appear  (§  6)  that  algebra  abbreviates  and 
greatly  simplifies  the  solution  of  certain  kinds  of  problems.  It 
will  also  be  shown  that  the  meaning  hitherto  attached  to  num- 
ber, as  well  as  its  mode  of  representation,  is  greatly  extended  in 
algebra;  and  that  the  "equation,"  which  plays  a  very  minor  part 
in  arithmetic,  is  of  great  importance  in  algebraic  investigations. 

2.  Number.  The  first  numbers  that  present  themselves  are 
those  which  arise  from  counting  and  from  measuring  things;* 
they  are  usually  called  whole  numbers,  and  also  integers,  but  may 
quite  appropriately  be  called  the  natural  numbers.  These  numbers 
are  always  definite,  and  are  represented  by  one  or  more  of  the 
Arabic  characters  0,  1,  2,  3,  4,  5,  6,  7,  8,  and  9. 

Out  of  combinations  of  these  natural  numbers  have  grown  other 
kinds  of  numbers,  such  as  fractions,  which  have  already  been 
studied  in  arithmetic,  and  still  other  kinds  which  will  "be  pre- 
sented in  later  chapters  of  this  book. 

*  Numbers  themselves  are  not  found  ready  made  in  nature ;  there  are,  how- 
ever, everywhere  things,  and  the  counting  or  the  measuring  of  these  gives  rise  to 
numbers.  Since  much  of  the  intercourse  of  life  is  concerned  with  the  things 
about  us,  and  with  their  relations  to  one  another,  and  since  these  relations  are 
expressed  by  means  of  numbers,  it  is  for  this  reason  alone — to  say  nothing  of 
other  excellent  reasons  —  of  fundamental  importance  that  numbers  and  their 
combinations  be  carefully  studied.  It  will  be  found  advantageous,  and  will 
add  clearness  of  view,  if  in  our  reasoning  about  numbers  we  frequently  go  back 
to  the  things  themselves  from  which  these  numbers  may  have  arisen. 

1 


2  ELEMENTARY  ALGEBRA  [Ch.  I 

3.  Arithmetical  processes,  (i)  Addition.  Fundamentally,  addi- 
tion of  natural  numbers  is  merely  counting. 

E.g.,  to  add  4  to  7,  means  to  find  that  number  which  is  four  greater  than 
seven ;  we  begin  therefore  with  7  and  count  four,  forward,  which  gives  11. 
Similarly  in  general. 

The  sign  of  addition  is  an  upright  cross  (+),  which  is  read  plus 
(meaning  more) ;  when  written  between  two  numbers,  it  means 
that  the  second  is  to  be  added  to  the  first. 

E.g.,  7  +  4  is  read  "  seven  plus  four,"  and  means  that  4  is  to  be  added  to  7. 

The  result  of  adding  two  or  more  numbers  is  called  their  sum; 
the  numbers  to  be  added  are  called  the  summands. 

It  is  evident  that  addition,  in  the  case  of  natural  numbers,  is  always  a  possible 
arithmetical  operation  ;  that  this  is  not  true  of  subtraction  will  be  seen  in  (ii) 
below. 

Two  short  parallel  horizontal  lines  (=)  are  used  to  express 
that  one  of  two  numbers  is  equal  to,  i.e.,  is  the  same  as,  the 
other ;  e.g.,  7  +  4  =  11.  This  expression  is  called  an  equation, 
and  is  read  "  seven  plus  four  equals  eleven." 

(ii)  Subtraction.  Subtraction  is  tlie  inverse*  of  addition; 
with  natural  numbers  it  is  a  counting  off. 

E.g.,  to  subtract  3  from  15,  we  begin  with  15  and  count  off  (or  backward) 

3  units,  thus:  14,  13,  12;  and  12  is  the  result  of  the  subtraction. 

In  other  words,  to  subtract  the  first  of  two  munbers  froiYi 
the  second  is  to  find  a  third  number  such  that  this  third 
number  plus  the  first  number  equals  the  second  number. 

The 'sign  of  subtraction  is  a  short  horizontal  line  (— ),  which  is 
read  minus  (meaning  less)  ;  when  written  between  two  numbers, 
this  sign  means  that  the  second  number  is  to  be  subtracted  from 
the  first. 

E.g.y  7  —  4  is  read  "  seven  minus  four,"  and  means  that  4  is  to  be  subtracted 
from  7. 


*  An  inverse  operation  may  be  defined  as  one  whose  effect  is  neutralized  by  the 
corresponding  direct  operation.  Addition  and  multiplication  are  direct  opera- 
tions; their  inverses  are  subtraction  and  division. 


3]  INTEOBUCTION  '  3 

The  result  of  subtracting  one  number  from  another  is  called 
their  difference,  and  also  the  remainder;  the  number  which  is  sub- 
tracted is  called  the  subtrahend,  and  the  one  from  which  the 
subtraction  is  made  is  called  the  minuend. 

In  the  above  example,  7  is  the  minuend,  4  the  subtrahend,  and  3  the  remainder, 
all  of  which  is  expressed  by  the  equation  7—4  =  3,  which  is  read  "  seven  minus 
four  equals  three." 

From  the  above  definition  it  follows  that  subtraction  is  a  possible  arithmetical 
operation  only  when  the  minuend  is  at  least  as  great  as  the  subtrahend. 

(iii)  Multiplication  is  usually  defined  as  the  process  (or 
operation)  of  taking  one  of  two  numbers,  called  the  multiplicand, 
as  many  times  as  there  are  units  in  the  other,  which  is  called  the 
multiplier.  In  this  sense  multiplication  is,  fundamentally,  the 
same  as  addition. 

E.ff.,  8  multiplied  by  5  means  that  8  is  to  be  used  5  times  as  a  summand;  i.e., 
the  product  of  8  multiplied  by  5  is  8+8  +  8  +  8  +  8. 

The  sign  of  multiplication  is  an  oblique  cross  (  x  ),  which  is  read 
multiplied  by ;  when  written  between  two  numbers,  it  means  that 
the  first  is  to  be  multiplied  by  the  second.  The  result  of  multi- 
plying one  number  by  another  is  called  their  product. 

Note.  The  definition  of  multiplication  just  given  applies  only  when  the  mul- 
tiplier is  an  integer.  Under  it,  multiplication  by  a  fraction  or  by  a  mixed  number 
has,  strictly  speaking,  no  meaning.  For  example,  let  it  be  required  to  multiply 
8by  5|;  to  do  this  under  the  definition  just  given,  it  is  necessary  to  take  8  as 
many  times  as  there  are  units  in  5|,  but  manifestly,  while  8  may  be  taken  addi- 
tively  five  times,  it  can  not  be  taken  tioo  thirds  of  a  time*  and  the  proposed 
problem,  therefore,  does  not  admit  of  solution  under  this  definition. 

A  far  more  useful  definition  of  multiplication  than  that  given 
above,  and  one  that  will  serve  all  future  needs,  may  be  stated  thus : 

The  product  of  two  numhers  is  the  result  obtained  hy 
performing  upon  the  first  of  these  numhers  {the  multi- 
plicand) tl%e  same  operation  that  must  he  perfonned  upon 
the  unit  to  obtain  the  second  {th^,  inultiplier) . 

This  definition  not  only  includes  the  former  one,  but  it  also 
gives  an  intelligible  meaning  to  multiplication  when  the  multiplier 
is  a  fraction  or  a  mixed  number. 

*  This  is  as  meaningless  as  "  to  fire  a  gun  two- thirds  of  a  time." 


4  ELEMENTARY  ALGEBRA  [Ch.  I 

E.g.,  consider  again  the  question  of  multiplying  8  by  5f ;  the  multiplier,  5|,  is 
obtained  from  the  unit  by  taking  the  unit  five  times,  and  ^  of  the  unit  twice,  as 
summands ; 
i.e.,  5|  =  l  +  l  +  l  +  l  +  l  +  i  +  i 

and,  therefore,  by  this  new  definition  of  multiplication, 

8x51  =  8  +  8  +  8  +  8  +  8  +  1  +  ! 
=  40  +  J#  =  45i 

(iv)  Division.  In  algebra  as  in  arithmetic,  to  divide  one  of 
two  given  numbers  by  another  is  to  find  a  number  which,  being 
multiplied  by  the  second  of  the  given  numbers,  will  produce  the 
first ;  the  symbol  of  division  is  -h,  and  is  read  divided  by. 

E.g.,  15  -^  5  =  3,  because  3  X  5  =  15 ;  the  first  of  these  equations  is  read  "  fifteen 
divided  by  five  equals  three." 

The  operation  of  dividing  one  number  by  another  is  called 

division,  the  first  of  the  given  numbers  is  called  the  dividend,  the 

second  is  the  divisor,  and  the  result,  i.e.,  the  number  sought,  is  the 

quotient. 

E.g.,  in  15  ^  6  =  3  the  dividend,  divisor,  and  quotient  are  15,  5,  and  3,  respec- 
tively. 

Note  1.  Observe  that,  under  the  above  definition,  the  test  of  the  correctness 
of  a  quotient  is  quotient  x  divisor  =  dividend. 

Division  is  therefore  the  inverse  of  multiplication  (cf.  footnote,  p.  2). 

NoTJB  2.  Observe  also  that  while  the  sum,  the  difference,  and  also  the  product 
of  any  two  integers  is  an  integer,  their  quotient  may  or  may  not  be  an  integer ; 
for  instance,  6  -f-  3  is  an  integer,  but  7-^3  and  5  -7-9  are  called  fractions  [cf. 
§  7  (V)]. 

4.  Symbols  of  continuation  and  deduction.  The  symbol  of  con- 
tinuation is  •••;  it  is  read  "and  so  on,"  or  "and  so  on  to,"  and 
is  used  to  denote  that  a  given  succession  of  numbers  is  to  con- 
tinue, either  without  end  or  up  to  a  given  number. 

E.g.,  1,  2,  3,  •••  is  read  "one,  two,  three,  and  so  on" ;  while  1,  2,  3,  •••  27  is 
read  "one,  two,  three,  and  so  on  to  twenty-seven." 

The  symbols  of  deduction  are  •••  and  .-. ,  and  are  read  "  since  "  and 
"  therefore,"  respectively. 

E.g.,  ■.•3x5  =  15,  .-.  15  -4-  5  =  3;  this  expression  is  read  "  since  three  multi- 
plied by  five  equals  fifteen,  therefore  fifteen  divided  by  five  equals  three." 

The  symbols  explained  in  this  section  are,  like  all  other  signs 
and  symbols,  merely,  abbreviations  for  longer,  expressions. 


3-5]  INTBODUCTION 


EXERCISES 

Read  the  following  expressions,  and  give  the  names  of  their  parts : 

1.  3  +  7  =  10.  3.   15  -  3  =  5. 

2.  13  -  8  =  5.  4.     4  X  6  =  24. 

5.  State  the  definitions  of  the  operations  indicated  in  exercises  1-4. 
Show  that  your  definition  of  multiplication  applies  also  to  cases  in  which 
the  multiplier  is  a  fraction  or  a  mixed  number. 

6.  Which  of  the  operations  in  exercises  1-4  are  direct,  and  what  are 
their  respective  inverse  operations?     Explain  your  answer. 

7.  How  is  the  correctness  of  an  inverse  operation  to  be  tested?  Illus- 
trate your  answer  by  testing  the  correctness  of  15  -^  3  =  5. 

Read  the  following  expressions  : 

8.  •.•5x3=  15,  .•.  15  -  3  =  5.        9.   •.•  5  +  8  =  13,  .•.  13  -  8  =  5. 
10.   The  numbers  1,  3,  5,  •••  are  called  odd  numbers.     The  sum  of  the 

numbers  1,  3,  5,  •••  13  is  49. 

5.  Literal  notation.  The  Arabic  characters  of  arithmetic,  viz., 
0,  1,  2,  8,  '"  9,  and  also  the  signs  +,  — ,  x,  -^,  and  =,  are  all 
retained  in  algebra,  and  each  with  its  precise  arithmetical  mean- 
ing ;  but  algebra  also  frequently  employs  some  of  the  letters  of  the 
alphabet  to  stand  for,  or  represent,  numbers.* 

E.g.,  in  a  certain  problem  it  may  be  agreed  (possibly  merely  for  brevity)  to 
let  n  stand  for  a  particular  number,  say  786 ;  in  that  case  —  (i.e.,  one  half  of  n) 

would,  in  the  same  problem,  stand  for  393,  while  3  n  {i.e.,n  +  n  +  n)  would  stand 
for  2358,  etc.  In  another  problem,  however,  n  may  be  employed  to  represent  any 
other  desired  number. 

One  advantage  of  representing  numbers  by  letters  is  explained  in  §  6  below; 
others  will  appear  later.  For  the  present  it  is  perhaps  sufficient  to  say  that,  just 
as  in  arithmetic  we  speak  of  4  books,  7  bicycles,  85  pounds,  3  men,  etc.,  so  in 
algebra  we  shall  frequently,  in  addition  to  these  expressions,  use  such  expressions 
as  a  books,  n  bicycles,  x  pounds,  y  men,  etc. 

When  it  is  necessary  to  distinguish  between  numbers  which 
are  represented  by  the  Arabic  characters  0,  1,  2,  •••,  and  numbers 
which  are  represented  by  letters,  the  latter  will  be  called  literal 
numbers. 

*  This  way  of  representing  numbers  is,  however,  not  entirely  new  to  the  stu- 
dent because,  even  in  arithmetic,  in  problems  concerning  "  interest,"  the  princi- 
pal, amount,  rate,  interest,  and  time  are  often  represented  by  the  letters  p,  a,  r, 
i,  and  t,  respectively. 


6  ELEMENTARY  ALGEBRA 


[Ch.  I 


The  properties  of  numbers  are,  of  course,  precisely  the  same 
whether  these  numbers  are  represented  by  the  Arabic  characters, 
by  letters,  by  words,  or  in  any  other  way. 

E.g.,  just  as  3  books +  8  books  =  11  books,  so  m  books +  ?i  books  =  (m+n) 

books ;  and  if  k  stands  for  20,  then  Sk-i ■2yfc==25. 

4 
Again,  just  as  7  —  3  means  that  3  units  are  to  be  subtracted  from  7  units,  so 
a~b  means  that  b  units  are  to  be  subtracted  from  a  units. 


EXERCISES 

1.  If  5  represents  16,  what  number  is  represented  by  2  s?    by  |  of  s, 

i.e.,  by  i?  by2s  +  -?* 
4  4 

2.  If  a,  b,  and  c  represent,  respectively,  2,  5,  and  8,  what  is  the  value 
of  3  a- 6?  of  a  +  &  +  c?   of  ^^^^? 

3.  If  a:  represents  the  number  of  panes  of  glass  in  a  window,  how 
may  the  number  of  panes  of  glass  in  3  such  windows  be  repi-esented  ? 

—  _  4.  If  a  suit  of  clothes  costs  8  times  as  much  as  a  hat,  and  if  d  stands 
for  the  number  of  dollars  which  the  hat  costs,  what  will  represent  the 
cost  of  the  suit?  How  may  the  combined  cost  of  the  suit  and  hat  be 
represented  ? 

5.  Since  ^  of  any  number  is  the  same  as  /g  of  that  number,  and  i  of 
a  number  is  the  same  as  y\  of  that  number,  what  is  the  remainder  when 
^  of  n  is  subtracted  from  |  of  n,  where  n  represents  any  number  what- 
ever? i.e.,  ^  -  ^  =  ? 

'  3      4 

6.  Just  as  37  may  be  represented  by  10  x  3  +  7,  so  10  ^  +  w  represents 
a  number  whose  tens'  digit  is  t  and  whose  units'  digit  is  u.  If  the  units', 
tens',  and  hundreds'  digits  of  a  number  are  represented  by  x,  y,  and  z, 
respectively,  how  may  the  number  itself  be  represented? 

7.  If  X  represents  the  number  of  years  in  a  man's  present  age,  how 
may  his  age  5  years  ago  be  represented  ?  What  will  represent  his  age 
12  years  hence? 

8.  If  X  represents  any  integer,  how  may  the  next  higher  integer  be 
represented?  The  next  above  that?  If  n  represents  any  integer,  does 
2  n  represent  an  even  or  an  odd  number?  How  may  the  next  higher  even 
number  be  represented?  Show  that  2w  —  3,  2n— 1,  2n  +  l,  2n  +  3,  •••, 
represent  consecutive  odd  numbers. 

*  In  these  exercises,  and  throughout  the  first  five  chapters  of  this  book,  a 
knowledge  of  the  ordinary  arithmetical  processes  is  assumed ;  the  fundamental 
principles  involved  will  be  studied  in  Chapter  VI. 


5-6]  INTRODUCTION  7 

9.  A  thermometer  reads  80*^  at  noon  and  falls  y°  during  the  next 
6  hours.     What  is  its  reading  at  6  o'clock? 

10.  What  number  multiplied  by  8  gives  the  product  40?  If  8  a:  =  40, 
what  is  the  value  of  a:?    li  3  y  +  b y  —  2  y  =  bi,  what  is  the  value  of  ?/ ? 

6.  One  advantage  of  literal  notation.  The  use  of  letters  to  repre- 
sent numbers  greatly  simplifies  the  solution  of  certain  kinds  of 
arithmetical  problems.  This  is  illustrated  in  the  examples  that 
follow. 

^  Prob.  1.     A  gentleman  paid  $45  for  a  suit  of  clothes  and  a  hat.    If 
the  clothes  cost  8  times  as  much  as  the  hat,  what  was  the  cost  of  each? 

Arithmetical  Solution 
The  hat  cost  "some  number  of  dollars,"  and  since  the  clothes  cost 
8  times  as  much  as  the  hat,  therefore  the  clothes  cost  8  times  "that  num- 
ber of  dollars,"  and  therefore  the  two  together  cost  9  times  "that  number 
of  dollars";  hence  9  times  "that  number  of  dollars"  is  $45,  therefore 
"that  number  of  dollars"  is  $5,  and  8  times  "that  number  of  dollars" 
is  $40;  i.e.,  the  hat  cost  $5,  and  the  clothes  cost  $40. 

This  solution  may  be  put  into  the  following  more  systematic 
form,  still  retaining  its  arithmetical  character. 

Some  number  of  dollars  =  the  cost  of  the  hat ; 

then      8  times  that  number  of  dollars  =  the  cost  of  the  clothes, 

9  times  that  number  of  dollars  =  the  cost  of  both, 
i.e.,        9  times  that  number  of  dollars  =  $45, 

that  number  of  dollars  =  $5,  the  cost  of  the  hat, 
and       8  times  that  number  of  dollars  =  $40,  the  cost  of  the  clothes. 

Algebraic  Solution 

The  solution  just  given  becomes  very  much  simplified  by  letting 
a  single  letter,  say  x,  stand  for  "some  number"  and  "that  num- 
ber "  which  occur  so  often  above ;  thus : 

Let  X  =  the  number  of  dollars*  the  hat  cost. 

Then  8  x  =  the  number  of  dollars  the  clothes  cost, 

and  X  +  8  a:  =  the  number  of  dollars  both  cost, 

i.e.f  9  a;  =  45, 

x=  5,  and  8  a:  =  40; 
i.e.,  the  hat  cost  $5,  and  the  clothes  cost  $40. 

*  The  letter  x  here  stands  for  a  number,  not  for  the  cost  of  the  hat ;  the  equa- 
tions are  numerical. 


8  ELEMENTARY  ALGEBRA  [Ch.  I 

Prob.  2.  Three  men,  A,  B,  and  C,  form  a  business  partnership  with  a 
capital  of  $30,000 ;  if  A  furnishes  twice  as  much  of  this  capital  as  B,  and 
C  furnishes  as  much  as  A  and  B  together,  how  much  does  each  furnish? 

Solution 
Let  X  =  the  number  of  dollars  furnished  by  B. 

Then  2  x  =  the  number  of  dollars  furnished  by  A, 

and  3  X  =  the  number  of  dollars  furnished  by  C ; 

and  the  algebraic  statement  of  the  conditions  of  the  problem  becomes 

a:  +  2x  +  3a;=  30,000, 
i.e.,  Qx  =  30,000, 

whence  x  =  5000,  2x  =  10,000,  and  Sx  =  15,000  ; 

i.e.,  A  furnishes  |10,000,  B  |5000,  and  C  |15,000  of  the  capital. 

Prob.  3.  Of  three  numbers  the  second  is  5  times,  and  the  third  2 
times,  the  first,  and  the  sum  of  these  numbers  exceeds  the  third  number 
by  42 ;  what  are  the  numbers  ? 

Solution 

Let  X  =  the  first  of  the  three  numbers. 

Then  5  x  =  the  second  of  the  three  numbers, 

and  2  X  =  the  third  of  the  three  numbers ; 

and  the  algebraic  statement  of  the  conditions  of  the  problem  becomes 

x  +  5x-h2x  =  2x  +  i^2, 
i.e.,  8  a:  =  2  a:  +  42, 

hence  6^=42,  TSubtract  2  x  from 

[_each  member 
therefore  x=7,  5  a:  =  35,  and  2  x  =  14 ; 

and  the  required  numbers  are,  respectively,  7,  35,  and  14. 

Note.  Observe  that  the  plan  of  each  of  the  foregoing  solutions  is  to  let  some 
letter,  say  x,  stand  for  one  of  the  unknown  jiumbers  (preferably  the  smallest), 
then  to  express  the  other  unknown  numbers  in  terms  of  x,  and  finally  to  trans- 
late into  algebraic  language  the  conditions  which  are  verbally  stated  in  the  prob- 
lem ;  this  last  statement  is  an  equation,  and  from  it  the  required  numbers  are 
easily  found. 

Observe  also  that  while  the  above  problems  can  be  solved  by  arithmetical 
analysis,  the  algebraic  solution  is  much  simpler. 


6]  INTRODUCTION 


PROBLEMS 

4.  In  a  room  containing  45  pupils  there  are  twice  as  many  boys  as 
girls.     How  many  boys  are  there  in  the  room  ? 

5.  If  a  horse  and  saddle  together  cost  $  90,  and  the  horse  cost  5  times 
as  much  as  the  saddle,  how  nmch  did  each  cost  ? 

6.  In  a  business  enterprise,  the  combined  capital  of  A,  B,  and  C  is 
121,000.  A's  capital  is  twice  B's,  and  B's  is  twice  C's.  What  is  the 
capital  of  each  ? 

7.  The  difference  between  two  numbers  is  8,  and  their  sum  is  30. 
What  are  the  numbers? 

8.  Divide  98  into  three  parts  such  that  the  second  is  twice  the  first 
and  the  third  is  twice  the  second. 

9.  A  number,  plus  twice  itself,  plus  4  times  itself,  is  equal  to  56. 
What  is  the  number? 

10.  The  sum  of  three  numbers  is  160 ;  two  of  these  numbers  are  equal, 
and  the  third  is  twice  either  of  the  others.     Find  the  numbers. 

11.  In  a  fishing  party  consisting  of  4  boys,  2  of  the  boys  caught  the 
same  number  of  fish,  another  caught  2  more  than  this  number,  and 
another  1  less ;  if  the  total  number  of  fish  caught  was  29,  how  many  did 
each  catch? 

12.  If  a  locomotive  weighs  3  times  as  much  as  a  car,  and  the  difference 
between  their  weights  is  50  tons,  what  does  the  locomotive  weigh  ? 

13.  Of  two  numbers,  twice  the  first  is  seven  times  the  second,  and 
their  difference  is  75 ;  find  the  numbers. 

Suggestion.    Let  1  x=  the  first  number,  then  2x  =  the  second. 

14.  An  estate  of  $  19,600  was  so  divided  between  two  heirs  that  5  times 
what  one  received  was  equal  to  9  times  what  the  other  received;  what 
was  the  share  of  each  ? 

15.  A  horse,  harness,  and  carriage  together  cost  |340;  if  the  horse 
cost  3  times  as  much  as  the  harness,  and  the  carriage  cost  \\  times  as 
much  as  the  horse  and  harness  together,  what  was  the  cost  of  each  ? 

16.  A,  B,  C,  and  D  together  buy  %  16,000  worth  of  railroad  stock.  B 
buys  three  times  as  much  as  A,  C  twice  as  much  as  A  and  B  together, 
and  D  one  third  as  much  as  A,  B,  and  C  together.     How  much  does  each 


10  ELEMENTARY  ALGEBRA  [Ch.  1 

17.  What  number  added  to  I  of  itself  equals  20  ? 

Solution 
Let  X  =  the  number. 

Then  x  +  ^  x  =  20, 

i.e.,  |x  =  20, 

.-.  '       cc  =  20-^1  =  15. 

18.  If  I  of  a  number  is  added  to  the  number,  the  sum  is  120;  what  is 
the  nural)er? 

19.  If  I  of  a  number  is  added  to  twice  the  number,  the  sum  is  35; 
what  is  the  number? 

20.  'I'he  difference  between  4  times  a  certain  number  and  |  of  that 
number  is  30;  what  is  the  number  ? 

21.  Three  times  A's  age  is  four  times  B's,  and  the  sum  of  their  ages 
exceeds  |  of  A's  age  by  24  years;  what  is  the  difference  between  their 
ages? 

22.  A  merchant  owes  a  certain  sum  of  money  to  A,  |  as  much  to 
B,  and  twice  as  much  to  C  as  he  owes  A;  various  persons  owe  him 
12  times  as  much  as  he  owes  B,  and  if  all  these  debts  were  paid,  the  mer- 
chant would  have  $4000.  What  is  the  total  amount  that  the  merchant 
owes  ? 

23.  A  boy  found  that  he  had  the  same  number  of  5,  10,  and  25  cent 
pieces,  and  that  the  total  amount  of  his  money  was  $3.20;  how  many 
coins  of  each  kind  had  h3? 

24.  Of  a  family  of  seven  children  each  child  is  2  years  older  than  the 
next  younger;  if  the  sum  of  their  ages  is  81  years,  how  old  is  the 
youngest  child? 

25.  In  a  number  consisting  of  two  digits,  the  digit  in  units'  place  is 
3  times  that  in  tens'  place,  and  if  these  digits  be  interchanged,  the  num- 
ber will  be  increased  by  36 ;  w^hat  is  tlie  number  (cf.  Ex.  6,  §  5)  ? 

26.  The  president  of  a  stock  company  owns  3  times  as  many  shares  as 
the  vice  president,  and  the  secretary  owns  6  shares  less  than  the  vice  presi- 
dent ;  if  these  three  men  together  own  539  shares,  how  many  shares  does 
each  own? 

27.  Three  newsboys  sold  a  total  of  191  papers  in  an  afternoon;  if  the 
second  sold  5  more  than  twice  as  many  as  the  first,  and  the  third  sold 
three  times  as  many  as  the  second,  how  many  did  each  sell? 

28.  A  tree,  whose  height  was  150  feet,  was  broken  off  by  the  wind, 
and  it  is  found  that  3  times  the  length  of  the  part  left  st-anding  is  the 
same  as  7  times  that  of  the  part  broken  off ;  how  long  is  each  part  ? 


i 


]  •  INTRODUCTION  11 

29.  In  a  yachting  party  consisting  of  36  persons,  the  number  of  chil- 
dren is  3  times  the  number  of  men,  and  the  number  of  women  is  one  half 
that  of  the  men  and  children  combined;  how  many  women  are  there  in 
this  party  ? 

30.  If  two  boys  together  solved  65  problems,  and  if  8  times  the  num- 
ber solved  by  the  first  boy  equals  5  times  the  number  solved  by  the  second 
boy,  how  many  did  each  boy  solve  ? 

31.  An  estate  valued  at  $  24,780  is  to  be  divided  among  a  family  con- 
sisting of  the  mother,  2  sons,  and  3  daughters ;  if  the  daughters  are  to 
receive  equal  shares,  each  son  twice  as  much  as  a  daughter,  and  the 
mother  twice  as  much  as  all  the  children  together,  what  will  be  the  share 
of  each? 

32.  A  library  contains  17  times  as  many  scientific  books,  and  6  times 
as  many  historical  books,  as  books  of  fiction;  if  the  books  of  fiction 
number  220  less  than  the  scientific  and  historical  books  together,  how 
many  books  are  there  in  this  library  ? 

33.  A,  B,  and  C  enter  into  a  business  partnership  in  which  A  furnishes 
6  times  as  much  capital  as  C,  and  B  furnishes  |  as  much  as  A  and  C 
together;  if  the  total  capital  is  |13,500,  how  much  is  furnished  by  each 
partner  ? 

7.  Operations  with  literal  numbers.  As  is  pointed  out  in  §  5,  the 
reasoning  employed  with  numbers  represented  by  letters  is  pre- 
cisely the  same  as  if  those  numbers  were  represented  by  the  Arabic 
characters.  It  may  be  worth  while,  however,  to  examine  the  fun- 
damental operations  a  little  more  closely. 

(i)  AdditioTi.  Just  as  3  +  7  means  that  7  is  to  be  added  to  3, 
so  too,  if  a  and  h  stand  for  any  two  numbers  whatever,  a  +  h 
means  that  h  is  to  be  added  to  a. 

Similarly,  a-\-x-{-p  means  that  x  is  to  be  added  to  a,  and  that 
p  is  then  to  be  added  to  that  sum ;  and  so  in  general. 

(ii)  Subtraction.  Just  as  15  —  9  means  that  9  is  to  be  sub- 
tracted from  15,  %o  x  —  y  means  that  y  is  to  be  subtracted  from  a;, 
whatever  the  numbers  represented  by  x  and  y. 

Note.  Observe  that,  while  addition  is  always  possible,  the  indicated  subtrac- 
tion a;  —  ?/  is  arithmetically  possible  only  when  the  number  represented  by  x  is  at 
least  as  great  as  that  represented  by  ?/. 

This  restriction  upon  the  relative  values  of  the  two  numbers  in  such  an  expres- 
sion as  x  — ?/  is  often  very  inconvenient;  in  Chapter  II  the  meaning  of  number  is 
so  extended  as  to  make  this  subtraction  possible  even  when  y  is  greater  than  x. 


12  ELEMENTARY  ALGEBRA  [Ch.  I 

(iii)  Multiplication.  Just  as  6  x  5  means  that  6  is  to  be 
multiplied  by  5,  so  6x3  means  that  h  is  to  be  multiplied  by  3. 
Again,  a  x  y  X  n  means  that  a  is  to  be  multiplied  by  y,  and  that 
their  product  is  then  to  be  multiplied  by  n ;  and  so  in  other  cases. 

Instead  of  the  oblique  cross  ( x ),  a  center  point  (•)  placed  be- 
tween two  numbers  (a  little  above  the  line  to  distinguish  it  from 
a  decimal  point)  is  frequently  used  as  a  sign  of  multiplication. 

E.g.,  instead  of  4x6,  3xn,  axk,  etc.,  it  is  usual  to  write  4  •  6,  3  •  n,  a  •  A;,  etc. 

And  even  the  center  point  is  usually  omitted  in  cases  where  its 

omission  causes  no  misunderstanding. 

E.g.,  3  Xn  =  3-n  =  3n,  and  aX  k  =  a  -  k=  ak;  but,  while  4  X  6  =  4  •  6,  it  can 
not  be  written  "46,"  for  in  that  case  it  would  be  confused  with  40  +  6. 

(iv)  Powers,  exponents,  etc.  Products  in  which  all  the  fac- 
tors are  identical  with  one  another  are  usually  written  in  an  abbre- 
viated form.  This  form  consists  of  the  repeated  factor  written 
only  once  and  having  attached  to  it  (at  the  right  and  slightly 
above)  the  number  which  tells  how  many  times  the  given  factor 
is  to  be  repeated. 

E.g.,  2  •  2  •  2  is  written  2^,  a  '  a  '  a  •  a  '  a  is  written  a^,  and  the  product  of  n 
factors  each  of  which  is  a  is  written  x^. 

The  expression  a?"  is  called  the  nth  power  of  x,  and  is  usually 
read  "  x  nth. " ;  the  number  n  is  called  the  exponent  of  the  power, 
and  X  is  called  the  base.  In  particular,  2^  is  the  third  power  of 
2,  the  exponent  is  3,  and  the  base  is  2. 

A  power  is  called  odd  or  even  according  as  its  exponent  is  odd 
or  even. 

Similarly,  a  product  in  which  the  factor  2  is  repeated  3  times, 
and  the  factor  5  is  repeated  2  times,  is  written  2''  •  5^.  And,  more 
generally,  the  expression  a^'b^'c^  is  the  product  of  a  repeated  m 
times,  b  repeated  n  times,  and  c  repeated  p  times;  it  is  read  "the 
mth  power  of  a,  multiplied  by  the  nth.  power  of  b,  multiplied  by 
the  pth  power  of  c." 

Note.  Under  the  definition  of  a  power  given  above,  it  is  evident  that  a^  has 
the  same  meaning  as  a,  and  the  exponent  1,  therefore,  need  not  be  written. 

The  second  and  third  powers  of  numbers  are,  for  geometric  reasons,  often  called 
by  the  special  names  of  square  and  cube,  respectively;  thus,  a!^  is  known  as  the 
"second  power  of  a,"  the  "square  of  a,"  and  also  as  "a  squared";  and  x^  is 
known  as  the  "  third  power  of  x,"  the  "  cube  of  x,"  and  also  as  "  x  cubed."  Cor- 
responding to  the  other  powers  there  are  no  such  special  names. 


7-8]  INTRODUCTION  13 

(v)  Division.  Just  as  40  -h  5  indicates  that  40  is  to  be  divided 
hj  5,  so  a-v-b  indicates  that  a  is  to  be  divided  by  b,  whatever  the 
numbers  represented  by  a  and  b ;  that  is,  (a-i-b)  •b  =  a  for  all 
values  of  a  and  b  [cf.  §  3  (iv),  note  1]. 

Other  forms  of  writing  a-i-b  are :  -,  a:b,  and   a/b. 

b 

In  algebra,  as  in  arithmetic,  if  the  divisor  is  not  exactly 
contained  in  the  dividend,  the  indicated  division  is  called  a 
fraction.* 

^'9',  I,  — .  -»  and  ^-±^  are  fractions. 
3     5     n  y 

It  is  to  be  remarked,  in  passing,  that  literal  numbers  may  be 
fractional  in  form  and  yet  have  integral  values,  and  vice  versa. 

E.g.,  — ,  though  fractional  in  form,  has  the  integral  value  3  if  a  =  12  and  6  =  4; 
b 
and  m  +  3  7i,  though  integral  in  form,  has  the  value  j|  if  m  =  j  and  n  =  j. 

8.  The  order  in  which  arithmetical  operations  are  to  be  performed. 
Signs  of  aggregation.  When  there  is  no  express  statement  to  the 
contrary,  a  succession  of  multiplications  and  divisions  is  under- 
stood to  mean  that  these  operations  are  to  be  performed  in  the 
order  in  which  they  are  written  from  left  to  right.  The  same 
rule  applies  in  the  case  of  a  succession  of  additions  and  sub- 
tractions. 

E.g.,  9  •  8  -^  6  •  2  means  that  9  is  to  he  multiplied  hy  8,  that  product  to  be  divided 
by  6,  and  the  resulting  quotient  to  be  multiplied  by  2;  it  does  not  moan  that  the 
product  of  9  by  8  is  to  be  divided  by  the  product  of  6  by  2 :  the  result  is  24,  and 
note. 

So,  too,  7  +  9  —  6  +  3  means  that  9  is  to  be  added  to  7,  then  6  subtracted  from 
that  sum,  and  finally  3  added  to  this  remainder ;  it  does  not  mean  that  6  +  3  is  to 
be  subtracted  from  7  +  9 :  the  result  is  13,  and  not  7. 

Again,  by  a  succession  of  the  operations  of  addition;  subtraction, 
multiplication,  and  division,  when  the  contrary  is  not  expressly 
stated,  it  is  customary  to  mean  that  all  the  operations  of  multi- 
plication and  division  are  to  be  performed  in  the  order  in  which 


*  A  fraction  is  usually  defined  as  "  one  or  more  of  the  equal  parts  into  w^hich  a 
unit  has  been  divided,"  but  this  definition  is  only  a  special  case  of  the  one  given 
above ;  it  is  meaningless  when  the  denominator  is  not  an  integer. 


14  ELEMENTARY  ALGEBRA  [Ch.  I 

they  are  written  from  left  to  right,  before  any  of  those  of  addi- 
tion and  subtraction  are  performed ;  the  resulting  expression  will 
then  contain  only  the  operations  of  addition  and  subtraction,  and 
these  operations  are  then  to  be  performed  in  the  order  in  which 
they  occur. 

E.g.,  the  expression  2  +  6-5  —  8-^2  means  2  +  30  —  4,  which  is  28. 

Should  the  writer  of  such  an  expression  desire  that  a  different 
meaning  be  given  to  the  expression  (e.g.,  that  one  or  more  of  the 
additions  and  subtractions  be  performed  before  some  of  the  mul- 
tiplications and  divisions  are  performed),  he  would  indicate 
his  meaning  by  employing  one  or  more  of  the  so-called  signs  of 
aggregation;  among  these  are  the  parenthesis  (  ),  the  brace  |  \,  the 
bracket  [  ],  and  the  vinculum  '.     An  expression,  included  in 

the  parenthesis,  brace,  or  bracket,  or  under  the  vinculum,  is  to  be 
regarded  as  a  whole,  and  is  to  be  treated  as  though  it  were  repre- 
sented by  a  single  symbol. 

E.f/.,  (2  +  6)  .5-4-3-(7  +  8-^2)=8-5-=-3  — 11,  i.e.,  2^.  So,  too,  (4  +  6)-^2  =  5. 
while  without  the  parenthesis  its  value  would  be  7. 

It  may  even  be  useful  sometimes  to  employ  one  sign  of  aggre- 
gation within  another. 

E.g.,  72 ^  {252 -  (24  •  4  +  6)}. 

In  such  a  case  the  innermost  sign  of  aggregation  is,  of  course,  to  be  attended  to 
first ;  the  value  of  the  above  expression  is  6. 


EXERCISES 

Find  the  value  of  each  of  the  following  expressions : 

1.   38-6  +  14-12-2.  2.   38  -  (6  +  14) -(12- 2). 


3.  9.  6 -4(36 -3 -2) +54 -(17 -12 -5). 

4.  12  .  3  -  (9  +  3  -  G)  .  18  -  6^=^. 

5.  {4  .  9  -  16  ^  2  -  (12  -  8)  ^  (4  +  6  -^  3)}  -  (6  -  2). 

6.  Give  a  definition  of  a  fraction  that  will  include  cases  in  which  the 
denominator  is  such  a  number  as  3|. 

7.  May  an  expression  be  fractional  in  /orm,  but  integral  in  value? 
Give  three  examples  of  this  kind. 


8-9]  INTRODUCTION  15 

Read  each  of  the  following  expressions,  then  tell  in  what  order  the 
indicated  operations  are  to  be  performed,  and  finally  find  the  numerical 
values  of  these  expressions  when  a  =  8,  6  =  3,  c  =  12,  and  d  =  \: 


8. 


12. 


c 

A 
ba 

'  d      hd 

ah 
cd 

9. 

4a 

+36-C-' 

10. 

(a  +  6)2- 

(a- 

by 

-4  aft. 

11. 

{ahc  +  h)  - 

-  (4  cc/  +  d)  -i 

-[^ 

-(a  +  4rf)], 

cd-. 

C2- 
rUd' 

h'^c 

13. 

.   a( 

;c-  6)  +  6(a 

o6c 
d 


(^l)' 


14.   {6  a  -  2  c  -  2  (/-'}  +  J^iii  -  (2  ft  •  ^/). 
4  c/3 

9.  Advantages  of  using  letters  to  represent  numbers.  Attention 
has  already  been  called  (§  6)  to  one  of  the  many  advantages  which 
result  from  the  use  of  letters  to  represent  numbers ;  two  further 
advantages  will  now  be  considered. 

(i)  Suppose  it  to  have  been  noticed,  in  a  few  particular  cases, 
that  half  the  sum  of  two  numbers  plus  half  their  difference  equals 
the  greater  of  these  numbers,  and  suppose  that  it  is  required  to 
ascertain  whether  this  is  true  for  a  certain  few  pairs  of  numbers 
only,  or  whether  it  is  true  for  all  possible  pairs  of  numbers. 

For  any  particular  pair  of  numbers  that  may  be  under  con- 
sideration, 15  and  7  for  example,  its  correctness  is  easily  verified, 

thus  AK\7       ip;       7 

but  after  having  made  this  verification  one  is  still  in  doubt  about 
every  untried  pair  of  numbers. 

If,  on  the  other  hand,  letters  are  employed,  it  may  be  proved, 
once  for  all,  that  the  above  property  belongs  to  every  pair  of 
numbers,  and  no  further  verifications  are  needed.  Thus,  let  a  and 
h  represent  any  two  numbers  whatever,  and  let  a  be  greater  than 
6;  then  x 

a4-5  ■  a  —  h_a     h  i^_^_^,^i^_^_^i^_^ 

2  2     "2      2     2      2~2      2      2      2~2     2~  ' 

which  proves  that  half  the  sum  of  any  two  numbers  ivJiatever,  plus 
half  their  difference,  equals  the  greater  of  these  numbers.  The 
literal  notation  has  here  served  to  prove  a  general  law. 


16  ELEMENTARY  ALGEBRA  [Ch.  I 

(ii)  Another  advantage  of  the  literal  notation  may  be  illustrated 
by  comparing  the  solutions  of  the  two  following  problems. 

Prdb.  1.  If  A  can  do  a  piece  of  work  in  15  days,  and  B  can  do  it  in 
10  days,  in  how  many  days  can  both  working  together  do  it? 

Prob.  2.  If  A  can  do  a  piece  of  work  in  a  days,  and  B  can  do  it  in  b 
days,  in  how  many  days  can  both  working  together  do  it? 

Solution  of  Problem  1 

Since  A  can  do  all  of  the  work  in  15  days,  therefore  he  can  do  j\  of  it 
in  one  day ;  similarly,  B  can  do  j\  of  it  in  one  day,  and  both  together  can 
therefore  do  ^^  +  Jj,  that  is,  I,  of  it  in  one  day ;  hence  it  will  take  both 
together  1  -f-  ^,  i.e.,  6,  days  to  do  the  work. 


Solution  of  Problem  2 

Since  A  can  do  the  work  in  a  days,  therefore  he  can  do  -  of  it  in 

1        '  " 

one  day ;   similarly,  B  can  do  -   of   the  work  in  one  day,   and  both 

together  can  do    -  +  7,  i.e.,   — ;— ,   of    it    in    one   day;    hence   it   will 
a     h  ah 

take  both  together  1  -^  ^       ,  that  is,     ^     .  days  to  do  the  work. 
ab  a  +  b 

The  reasoning  in  the  two  solutions  just  given  is  exactly  the 
same ;  it  is  to  be  observed,  however,  that  while  in  the  course  of 
the  first  solution  the  numbers  given  in  that  problem  (viz.,  15  and 
10)  have,  by  combining,  completely  lost  their  identity  before  the 
result  is  reached,  yet  the  numbers  given  in  the  second  problem 
(viz.,  a  and  b)  preserve  their  identity  to  the  end. 

Because  of  this  fact  the  answer  to  the.  second  problem  may  be 
used  as  2i,  formula  by  means  of  which  the  answer  to  any  other  like 
problem  may  be  immediately  written  down.  Thus,  if  a  =  15  and 
h  =  10,  then  the  second  problem  becomes  exactly  like  the  first, 

and  its  answer,  viz., ,  becomes  - — '- ,  which  is  6  as  before. 

a  +  h  15  +  10 

In  other  words,  the  solution  of  the  second  problem  includes  the 

solution  of   every  other   similar    problem  ;    numerical   problems 

like  the  first  are  merely  particular  cases  of  the  second. 


9-10]  INTBOBUCTION  17 

10.  Recapitulation.  Two  things  mentioned  in  this  chapter 
must  be  carefully  kept  in  mind  when  reading  the  following  pages; 
they  are :  (1)  the  somewhat  broader,  and  at  the  same  time  more 
precise,  definitions  *  of  the  fundamental  arithmetical  operations ; 
and  (2)  the  advantages  connected  with  the  use  of  letters  to  repre- 
sent numbers. 

While  the  Arabic  characters,  1,  2,  3,  •  •  •,  always  represent  the 
same  numbers,  wherever  they  occur,  a  letter  may  be  chosen  to 
represent  one  number  in  one  problem,  and  a  different  number  in 
another  problem ;  a  letter  may  also  represent  a  number  to  which 
no  specific  value  is  assigned  (cf.  §  9),  as  well  as  a  number  whose 
value  is  at  first  unknown  and  is  to  be  found  in  the  course  of  the 
solution  of  the  problem  (cf.  §  6). 


EXERCISES 

1.  Express  the  following  indicated  products  by  means  of  the  expo- 
nent notation  :  3  •  .3  •  3  •  3  •  3  ;  a  -  a  ■  a  •  a;  x  •  a:  •  a:  •••  to  12  factors ; 
5  •  5  •  5  ...  to  n  factors;  ax  >  ax  -  ax  '••  to  k  factors. 

2.  Define  the  expressions:  power,  base,  and  exponent,  and  illustrate 
your  meaning  by  means  of  exercise  1. . 

3.  Express  the  following  numbers  as  products  of  powers  of  prime 
numbers:  48,  200,  972,  and  1183. 

When  a  =  f  and  6  =  |,  verify  the  following  statements : 

4.  a(a  +  26)  =a2+2a&.  6.    (a  -  ft)3  =  a^  -  3  a2^,  +  3  ^^2  _  53. 

5.  (a  +  6)2  =  a2  4-  2  a&  +  6=.  7.    (a  +  6)  (a  -  &)  =  a^  -  h\ 

Find  the  numerical  value  of  each  of  the  following  expressions  when 
a  =  3,  &  =  I,  c  =  i,  a:  =  4,  y  =  2,  m  =  5,  and  n  =  2  : 

o    a^ft"  —  c^x  e.a-^-x^c-^mc'^ 

o. •  ». 1 

xy'^  —  ax**  0  ■{■  y      x  +n      n" 


*  These  definitions  pave  the  way  for  the  proofs  of  some  fundamental  laws  to  be 
given  later. 


CHAPTER   II 
POSITIVE  AND  NEGATIVE  NUMBERS 

11.  General  remarks.  As  already  pointed  out,  a.n  important  use 
of  numbers  is  to  enable  man  to  express,  in  a  brief  and  simple  way, 
the  relations  of  the  things  which  are  everywhere  round  about 
him.  At  first  he  used  only  the  natural  numbers,  i.e.,  the  integers, 
to  express  these  relations,  but  as  his  need  and  desire  for  precision 
and  conciseness  increased,  he  found  it  necessary  to  extend  his 
number  system  so  as  to  include  in  it,  not  only  fractions,  but  also 
other  kinds  of  numbers,  some  of  which  will  presently  be  studied. 

E.g.,  when  he  wished  to  express  even  so  simple  a  relation  as  that  between  the 
lengths  of  two  lines,  he  found  that  integers  alone  are  not  sufficient  unless  the 
lengths  of  these  lines  happen  to  be  such  that  the  longer  can  be  divided  into 
parts  each  of  which  will  be  just  as  long  as  the  shorter;  thus,  if  the  given  lines 
are  12  ft.  and  5  ft.  long,  respectively,  then  the  relation  between  their  lengths 
can  not  be  exactly  expressed  by  an  integer,  because  12  -^  5  is  not  an  integer. 

In  order  to  meet  this  and  other  like  needs,  man  extended  his  number  system 
so  as  to  make  the  arithmetical  operation  of  division  always  possible,  i.e.,  he 
included  common  fractions  in  his  number  system  (§  3,  note  2).  Before  fractions 
were  introduced,  division  was  possible  only  in  the  comparatively  few  cases  in 
which  the  dividend  happened  to  be  a  multiple  of  the  divisor. 

12.  Need  of  negative  numbers.  In  §  11  it  is  shown  that  a 
number  system  consisting  of  integers  only  is  not  sufficient  for 
man's  needs,  but  that  if  the  system  be  so  enlarged  as  to  make 
division  always  possible,  i.e.,  so  as  to  include  fractions  also,  this 
enlarged  system  will  serve  him  far  better  —  indeed  this  enlarged 
system  serves  all  the  purposes  of  ordinary  arithmetic. 

In  the  study  of  algebra,  however,  there  are  many  considera- 
tions which  make  it  very  advantageous  to  enlarge  the  number 
system  still  further. 

To  illustrate :  on  every  hand  there  are  found  things  which  stand  in  a  relation 
of  opposition  to  each  other  —  e.gf.,  assets  and  liabilities  in  business,  latitude  north 
and  latitude  south  of  the  equator,  temperature  above  zero  and  temperature 
below  zero,  etc.  — and  if  the  relations  between  these  opposite  things  are  to  be 
expressed  in  the  simplest  possible  way,  then  there  must  be  numbers  which  stand 
in  this  same  relation  of  opposition  to  each  other. 

18 


11-13]  POSITIVE  AND  NEGATIVE  NUMBERS  19 

How  to  enlarge  the  number  system  —  which  now  consists  of 
integers  and  fractions  (§  11)  —  so  that  it  will  meet  the  require- 
ments just  now  pointed  out,  becomes  evident  if  it  be  observed 
that  all  such  cases  of  opposition  as  those  mentioned  on-  the  pre- 
ceding page,  may  be  arrived  at  by  subtracting  a  number  from  one 
that  is  less  than  itself. 

E.g.,  if  a  business  man  whose  assets  are  ^5000  loses  ^6000,  i.e.,  if  $6000  be 
subtracted  from  his  $5000  of  assets,  it  leaves  him  not  only  without  any  assets, 
but  with  $  1000  of  liabilities,  i.e.,  he  has  $  1000  less  than  nothing;  if  from  latitude 
40°  north  50°  be  subtracted  (counted  off),  the  result  is  latitude  10°  south;  if  the 
thermometer  records  5°  above  zero  and  the  temperature  falls  8°,  it  will  then 
record  3°  below  zero ;  etc. 

Hence,  if  the  number  system  be  so  enlarged  as  to  make  subtrac- 
tion always  possible,  even  when  the  subtrahend  is  greater  than  the 
minuend,  this  enlarged  system  of  numbers  will  provide  for  all 
such  cases  of  opposition  as  those  above  mentioned.  The  nature 
of  these  new  numbers  will  be  more  closely  examined  in  the  next 
article. 

Note.  The  considerations  mentioned  in  §§  11  and  12  demand,  respectively, 
that  the  natural  number  system  be  extended  so  as  to  make  division  and  subtrac-' 
tion  always  possible,  i.e.,  so  as  to  give  a  meaning  to  the  expressions  a  -r-  6  and 
a—b,  whatever  the  relative  values  of  a  and  6. 

There  are,  however,  other  important  considerations  which  lead  to  the  same 
conclusions;  e.f/.,  algebra  makes  extensive  use  of  letters  to  represent  numbers, 
and  it  often  happens,  as  in  the  problems  of  §  0,  that  the  number  represented  by  a 
given  letter  may  be  unknown  until  after  the  problem  is  solved  ;  if  then  the  num- 
ber system  consists  of  integers  only,  and  if  a  and  b  represent  two  numbers  whose 
values  are  not  yet  known,  then,  should  the  combination  a  -r-  &  present  itself  in  a 
problem,  one  would  not  know  whether  or  not  it  could  be  treated  as  a  number 
(because  it  would  be  a  number  of  the  given  system  only  if  a  happened  to  be  a 
multiple  of  6),  and  further  progress  with  the  problem  must  necessarily  cease.  A 
much  wiser  plan  is,  of  course,  to  extend  the  number  system  so  as  to  make  a  -4-  6 
represent  a  number,  whatever  the  relative  values  of  a  and  6  {i.e.,  to  include  frac- 
tions in  the  number  system) ;  then  the  solution  may  be  continued  and  the  proper 
interpretation  given  at  the  end.  A  similar  argument  applies  to  such  an  expression 
as  a  — 6. 

13.  Negative  numbers  introduced.  The  natural  numbers  arranged 
in  a  series  increasing  by  one  from  left  to  right,  and  therefore 
decreasing  by  one  from  right  to  left,  are 

1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  ...; 


20  ELEMENTARY  ALGEBRA  [Ch.  II 

addition  is  performed  by  counting  toward  the  right  (cf.  §  3),  and 
subtraction  by  counting  toward  the  left,  in  this  series.  More- 
over, addition  is  always  possible  because  this  series  extends  with- 
out end  toward  the  right,  and  subtraction  is  arithmetically  possible 
only  when  the  subtrahend  is  not  greater  than  the  minuend  because 
this  series  is  limited  at  the  left. 

What  has  just  been  said  shows  that  to  make  subtraction  with 
natural  numbers  always  possible,  it  is  only  necessary  to  add  to 
the  present  number  system  such  numbers  as  will  continue  the 
above  series  indefinitely  toward  the  left. 

Let  the  result  of  subtracting  1  from  1  be  designated  by  0 ;  of 
subtracting  1  from  0,  by  ~1 ;  of  subtracting  1  from  "1,  by  ~2 ;  of 
subtracting  1  from  ~2,  by  ~3,  etc. ;  with  these  new  numbers  in- 
cluded, and  arranged  as  before,  the  series  becomes 

-.,  -6,  -5,  -4,  -3,  -2,  -1,  0,  1,  2,  3,  4,  5,  6,  7,  •.., 

which  extends  without  end  toward  the  left  as  well  as  toward  the 
right. 

Since  in  this  enlarged  series  each  number  is  less  by  one  than 
the  next  number  at  its  right  (and  therefore  greater  by  one  than 
the  next  number  at  its  left),  therefore  addition  and  subtraction 
with  natural  numbers  may,  as  before,  be  performed  by  counting 
toward  the  right  and  left  respectively. 

E.g.,  to  subtract  8  from  5,  i.e.,  to  find  the  number  which  is  8  less  than  5,  we 
begin  at  5  and  count  8  toward  the  left,  arriving  at  -3;  hence,  5  —  8  =  -3. 

Similarly,  4  —  6  =  -2,  4—  9  =  -5,  — 2  —  3  =  -5,  etc. ;  hence,  besides  indicating  a 
particular  place  in  the  enlarged  number  series,  -5  also  indicates  that  the  subtra- 
hend is  5  greater  than  the  minuend.*    Similarly  in  general. 

Again,  to  add  7  to  -4,  i.e.,  to  find  the  number  which  is  7  greater  than  —4,  we 
begin  at  -4  and  count  7  toward  the  right,  arriving  at  3.    Similarly  in  general. 

14.  Negative  numbers  defined.  Numbers  less  than  0  are  called 
negative  numbers,  and  are  written  thus:  ~1,  ~2,  ~3,  ••• ;  while  num- 
bers greater  than  0  are,  for  distinction,  called  positive  numbers, 

*  Such  an  expression  as  4  —  9  =  -5  is,  of  course,  not  to  be  understood  to  mean 
that  9  actual  units  of  any  kind  can  be  subtracted  from  4  such  units ;  4  of  the  9 
units  may  be  immediately  subtracted,  leaving  the  other  5  units  to  be  subtracted 
later  if  there  is  anything  from  which  to  subtract ;  in  this  sense  the  number  -5 
may  be  said  to  indicate  a  postponed  subtraction,  and  thus  to  have  a  suhtractive 
quality ;  hence  the  appropriateness  of  attaching  the  minus  sign  to  such  numbers. 


13-15]  POSITIVE  AND  NEGATIVE  NUMBERS  21 

and  are  written  either  ''"1,  +2,  "'"3,  •••,  or,  when  there  is  no  danger 
of  confusion,  simply  1,  2,  3,  •••. 

Positive  and  negative  numbers  taken  together  are  sometimes 
called  algebraic  numbers,  while  positive  numbers  alone  are  called 
arithmetical  numbers.  The  signs  ^  and  ~  employed  in  the  alge- 
braic numbers  above  are  called  signs  of  quality,  or  simply  the 
signs,  of  these  numbers.  Two  algebraic  numbers,  one  of  which 
is  positive  and  the  other  negative,  are  said  to  be  of  opposite 
quality,  or  to  have  unlike  signs,  while  if  both  numbers  are  positive, 
or  both  negative,  they  are  of  the  same  quality,  i.e.,  they  have  like 
signs.  A  number  written  without  a  sign  is  understood  to  be 
positive ;  the  negative  sign,  however,  is  never  omitted. 

The  numbers  ~1,  ~2,  ~3,  •••,  are  read:  negative  one,  negative  tivo, 
negative  three,  etc.,  and  also  minus  one,  minus  two,  etc. ;  and  the 
numbers  +1,  +2,  +3,  •••,  i.e.,  1,  2,  3,  •••,  are  read:  positive  one,  posi- 
tive two,  etc.,  also  plus  one,  plus  two,  plus  three,  etc.,  or  simply  one, 
two,  three,  etc. 

By  the  absolute  value  of  a  number  is  meant  its  mere  magnitude 
irrespective  of  its  quality ;  thus,  ~2  and  +2  have  the  same  abso- 
lute value,  so  too  in  general  have  ~a  and  """a,  whatever  the  number 
represented  by  a. 

Two  numbers  which  have  the  same  absolute  value,  but  which 
are  of  opposite  quality,  are  called  opposite  numbers ;  thus,  5  and 
~5  are  opposite  numbers,  so  too  are  "^a  and  ~a,  whatever  the 
number  represented  by  a. 

15.  Interpretation  of  negative  numbers.  The  interpretation  of  a 
negative  number  depends  upon  the  nature  of  the  problem  which 
gives  rise  to  it. 

E.g.,  a  lady  with  S15  in  her  purse  goes  shopping  and  makes  purchases 
amounting  to  $12  ;   how  much  money  has  she  left? 

Here  the  answer  is  clearly  15  — 12  dollars,  that  is,  3  dollars.  Had  the  pur- 
chases amounted  to  $  19,  the  answer  would  have  been  15  — 19  dollars,  that  is, 
-4  dollars  ;  and  the  -4  dollars  would  mean  that  she  not  only  had  no  money  left, 
but  that  she  was  4  dollars  in  debt. 

In  this  case  then,  when  possessions  are  under  consideration,  the  negative  num- 
ber means  indebtedness. 

The  student  should  now  re-read  §  12 ;  he  should  also  show  that 
if  in  a  certain  problem  temperature  above  zero  is  under  considera- 


22  ELEMENTARY  ALGEBRA  [Ch.  II 

tion,  then  a  negative  number  means  temperature  below  zero ;  simi- 
larly, if  positive  numbers  are  used  to  represent  degrees  of  north 
latitude,  then  negative  numbers  will  mean  degrees  of  south  lati- 
tude, etc. ;  in  other  words,  negative  numbers  must  in  all  cases  be 
interpreted  as  representing  things  opposite  in  character  to  those 
dealt  with  in  the  problem. 

EXERCISES 

[The  following  questions  should  be  supplemented  by  others  asked  by  the 
teacher.] 

1.  If  temperature  above  zero  be  regarded  as  positive,  interpret  the 
following  temperature  record  taken  from  a  U.  S.  Weather  Bureau  report : 
Albany,  +8"^;  Bismarck  (S.D.),  -11°;  Buffalo,  -2';  Chattanooga,  +26^; 
Denver,  "5° ;  Galveston,  +34'' ;   Marquette,  "9° ;   Oswego,  +1''. 

2.  How  much  warmer  is  it  at  Albany  than  at  Bismarck  in  the  above 
record?  at  Buffalo  than  at  Denver?  at  Buffalo  than  at  Chattanooga? 

3.  Answer  the  questions  in  Ex.  2  if  the  word  "colder"  be  put  in  place 
of  "  warmer." 

4.  The  value  of  all  the  available  property  of  a  merchant  is  a  dollars, 
and  his  total  indebtedness  is  b  dollars,  hence  the  value  of  his  estate  is 
(a  —  b)  dollars.  In  such  a  case  is  it  possible  that  h  is  greater  than  a? 
If  so,  what  kind  of  a  number  is  a  —  6?  In  this  case  how  should  this 
negative  number  be  interpreted?  Can  one  actually  pay  out  more  money 
than  he  has? 

5.  If  assets  are  represented  by  positive  numbers,  how  may  indebted- 
ness be  represented?  Interpret  the  financial  conditions  represented  by 
the  following  numbers:  $+783;  $"2568;  $'374.20;  and  1.(856  -  1232). 

Also  interpret  these  conditions  if  indebtedness  be  represented  by  posi- 
tive numbers. 

6.  A  boy  who  weighs  54  lb.  is  playing  with  a  toy  balloon  which  pulls 
upward  with  a  force  of  6  lb. ;  if  the  boy  were  weighed  while  holding  the 
balloon,  what  would  be  the  combined  weight?  If  +54  lb.  represents  the 
weight  of  the  boy,  what  w^ould  represent  the  tceight  of  the  balloon  ? 

7.  In  Ex.  6  the  combined  weight  of  the  boy  and  the  balloon  may  be 
represented  as  (+54  -f  "6)  lb.,  hence  adding  the  ne,2:ative  number  cancels 
part  of  the  positive  number ;  is  this  true  in  general  for  additions  of  posi- 
tive and  negative  rmmbers?    Illustrate  your  answer. 


15-16]  POSITIVE  AND  NEGATIVE  NUMBERS  23 

8.  If  distances  upstream  on  a  river  be  indicated  by  positive  numbers, 
what  would  "5  miles  along  this  stream  mean  V  Indicate  by  a  number 
and  sign  the  distance  and  direction  that  a  boat  would  Jioat  on  this  stream, 
in  1|  hours,  if  ttie  river  flows  2-J-  miles  an  hour. 

9.  An  oarsman  who  can  row  4  miles  an  hour  in  still  water  is  rowing 
upstream  on  the  river  in  Ex.  8 ;  show  tliat  the  distance  he  will  go  in  one 
hour  is  (4  4-  "2^)  miles.  Here  too  adding  a  negative  number  to  a  posi- 
tive number  cancels  it  in  part.     How  far  upstream  can  he  row  in  7  hours  ? 

10.  An  ocean  steamer  is  in  12°  east  longitude;  if  east  longitude  be 
indicated  by  positive  numbers,  and  if  the  vessel  moves  westward  through 
7°  of  longitude  per  day,  indicate  by  a  number  and  sign  the  longitude  of 
the  vessel  4  days  hence;  1|  days  hence;  2  days  ago. 

11.  If  the  vessel  in  Ex.  10  sails  westward  for  2  days  and  then,  being 
disabled,  drifts  1^°  eastward,  what  is  its  longitude? 

12.  What  is  meant  by  the  absolute  value  of  a  number  ?  Which  is  the 
greater,  8  or -12 ?  Why?*  Which  of  these  numbers  has  the  greater  ab- 
solute value? 

16.  Addition  of  negative  numbers.  In  order  to  understand  just 
what  is  meant  by  adding  a  negative  number  to  any  given  number, 
one  has  only  to  recall  the  essential  meaning  of  a  negative  num- 
ber. The  symbol  ~3,  for  example,  means  (and  may  always  be 
replaced  by)  a  subtraction  in  which  the  subtrahend  exceeds  the 
minuend  by  3  units,  i.e.,  it  is  equivalent  to  an  unperformed  (post- 
poned) subtraction  of  3  units. t  Hence,  to  add  ~3  to  any  number 
whatever  means  to  subtract  +3  from  that  number. 

E.g.,  8+-3  =  8  — 3  =  5;  4  + -10  =  4- 10  =-6;  -9  +  -5  = -9-5  = -14;  etc. 

Manifestly  the  above  reasoning  applies  to  any  negative  num- 
bers whatever,  hence  the  sum  of  two  or  more  negative  num- 
bers is  a  negative  number  whose  absolute  value  is  the  sum 
of  the  absolute  values  of  the  given  numbers ; 

And  the  suin  of  a  negative  and  a  positive  number  is  a 
number  whose  absolute  value  is  the  difference  of  the  abso- 
lute values  of  the  two  given  numbers,  and  whose  sign  is 
that  of  the  larger  of  these  numbers. 

*  Compare  §  117.  t  Compare  footnote,  p.  20. 


24  ELEMENTARY  ALGEBRA  [Ch.  II 

EXERCISES 

Find  the  value  of  each  of  the  following  expressions  : 

1.  13 +  -4.  3.    -6 +  10 +-7.  5.    -6t  +  10+-ll|.      ,     y 

2.  -8 +-3.  4.   3^  +  -9i+-5|.  6.   "2  + -13  +  8  + -4 +  6. 

7.  Regarding  a  negative  number  as  a  postponed  subtraction,  show 
that  the  result  in  Ex.  6,  and  in  all  others  like  it,  might  be  found  by 
adding  the  positive  numbers  separately,  and  the  negative  numbers  sepa- 
rately, and  then  uniting  these  two  sums. 

8.  If  money  in  hand,  or  to  be  received,  is  represented  by  a  positive 
number,  then  how  should  money  owed  (a  postponed  subtraction),  or  to  be 
paid  out,  be  represented  ? 

Indicate  by  a  sum  of  positive  and  negative  numbers  that  a  man  had 
$20  and  received  f  15  more,  and  that  he  paid  out  for  various  things  $8, 
$3,  and  $7.50;  also  show  in  two  ways  that  he  then  had  $16.50  left. 

9.  If  distances  westward  from  a  certain  point  be  indicated  by  posi- 
tive numbers,  how  should  distances  to  the  eastward  be  indicated? 

A  wheelman  after  riding  37  miles  westward  from  a  certain  point  rides 
back  12  miles;  show  that  37 +  "12  miles  indicates  both  his  direction  and 
distance  from  the  starting  point. 

10.  Indicate  by  a  sum  of  positive  and  negative  numbers  what  tempera- 
ture is  now  registered  by  a  thermometer  which  stood  at  4°  above  zero, 
then  rose  2°,  later  fell  9°,  and  then  rose  2i°  (cf.  Ex.  9). 

11.  Make  up  exercises  similar  to  8,  9,  and  10  to  illustrate  exercises  1-6 ; 
observe,  however,  that  the  demonstration  given  in  §  16  relies  wholly  upon 
the  definition  of  a  negative  number,  and  is  in  no  way  dependent  upon 
any  illustration. 

12.  From  the  reasoning  in  §  16  it  follows  that  in  adding  a  positive 
and  a  negative  number,  negative  units  and  positive  units  cancel  each 
other ;  show  that  this  is  true  in  aU  the  illustrations  above. 

17.  Subtraction  of  negative  numbers.  Since  subtraction  is  the 
inverse  of  addition,  i.e.,  since  to  subtract  any  number,  a,  from 
another  number,  6,  means  to  find  the  number  to  which  a  must  be 
added  to  produce  h,*  therefore  the  results  of  §  16  may  be  used  to 
show  how  to  subtract  negative  numbers. 

*  Definition  of  subtraction,  §  3  (iii). 


16-17]  POSITIVE  AND    NEGATIVE  NUMBERS  25 

Thus,  to  subtract  ~3  from  8  means  to  find  the  number  to  which 
-3  must  be  added  to  produce  8,  and  by  §  16  this  number  is  11, 
hence  8 -"3  =  11; 

but  8  +  3  =  11, 

8--3  =  8  +  3. 
Similarly,  15  -  "2  =  15  +  2;  4  -  "9  =  4  +  9;  "8  -"3  ="8  +  3; 

and,  in  general,     +a—~h=  +a  ++6,  and  -a—~h  —  ~a  ++&, 

whatever  the  numbers  represented  by  a  and  h  ;  i.e.,  subtracting 
a  negative  number  from  any  given  number  {positive  or 
negative)  gives  tl%e  same  result  as  adding  a  positive  num- 
ber of  the  same  absolute  value  to  the  given  number. 

Note.  If  three  or  more  algebraic  numbers  are  to  be  combined  by  addition  and 
subtraction,  the  order  in  which  these  operations  are  to  be  performed,  when  there 
is  no  express  indication  to  the  contrary  (parenthesis,  bracket,  etc.).  is  understood 
to  be  from  left  to  right  as  in  §  8.     E.g.,  +11  -+4  +-2  =+7  +-2  =+5. 

Moreover,  since  the  subtraction  of  an  algebraic  number  is  equivalent  to  the 
addition  of  its  opposite,  such  an  expression  as  +11— +4 +-2  (above)  is  usually 
spoken  of  as  an  algebraic  sum. 

EXERCISES 

1.  To  what  number  must  "5  be  added  to  produce  12  ?  What  then  is 
the  value  of  12  —  -5  ?  Answer  these  questions  if  12  is  replaced  by  3 ; 
by  -2  ;  by  a; ;  by  4  +  n. 

Find  the  value  of  each  of  the  following  expressions : 

2.  9 --6.  3.    -4 --12.  ■  4.   26§  - -41- 

5.  A  "  rule  "  for  subtracting  one  number  from  another  is  often  stated 
thus :  "  reverse  the  sign  of  the  subtrahend  and  proceed  as  in  addition." 
By  means  of  §  17  establish  the  correctness  of  this  rule  when  the  subtra- 
hend is  a  negative  number. 

6.  Using  positive  numbers  to  represent  money  in  hand  or  receivable, 
illustrate  the  fact  that  subtracting  a  negative  number  from  a  positive 
number  increases  that  number.  Does  subtracting  a  negative  number 
always  enlarge  the  minuend?    Is  it  so  in  -7  —"3? 

7.  In  the  extended  number  series  of  §  13,  viz.,  •••,  "3,  -2,  -1,  0,  1,  2,  3, 
4,  •••,  how  by  counting  may  we  add  5  to  3?  to  "2?  to -8?  Do  we 
count  forward  or  backward  when  adding  a  positive  integer?  Since  sub- 
traction is  the  inverse  of  addition,  which  way  should  we  count  when 
subtracting  a  positive  integer?  State  and  explain  the  corresponding 
facts  for  adding  and  subtracting  negative  integers. 


26  ELEMENTARY  ALGEBRA  [Ch.  II 

Simplify  each  of  the  following  expressions,  that  is,  find  the  value  of 
each  of  these  algebraic  sums  : 

8.  137 +-86 --7 +-26 -8.  10.  4p --54^ +-38|  -  28. 

9.  -54  +-864  +  732  -"413  -  36.       11.   18  --4'  -  13^  +"6  --17^. 

12.  Mount  Washington  is  6290  feet  above  the  sea  level.  Pikes  Peak 
is  14,083  feet  above  the  sea  level,  and  a  place  near  Haarlem,  in  Holland, 
is  16^  feet  below  the  sea  level.  Find  by  subtraction  how  much  higher 
Pikes  Peak  is  than  Mount  Washington ;  and  also  how  much  higher 
Mount  Washington  is  than  the  place  near  Haariem. 

13.  An  engineer  when  making  measurements  for  the  grade  of  a  street 
indicates  the  distances  of  points  above  a  certain  horizontal  reference  plane 
by  positive  numbers,  and  those  that  are  below  this  plane  by  negative 
numbers.  Show  that  the  difference  of  level  between  any  two  points  may 
always  be  found  by  subtraction.  Also  draw  figures  to  illustrate  several 
different  cases. 

18.  Product  of  two  algebraic  numbers.  Rule  of  signs.  The  prod- 
uct of  any  two  algebraic  numbers  is  readily  obtained  from  the 
definition  of  a  product,  which  is  given  in  §  3  (iii),  viz.,  the  product 
of  any  two  numbers  is  the  result  obtained  by  performing  upon 
the  multiplicand  the  same  operation  that  must  be  performed  upon 
the  positive  unit  to  get  the  multiplier. 

E.g.,  since  3  =  1  +  1  +  1, 

therefore  8- 3  =  8  +  8+8  =  24;' 

and  -8.3=-8+-8+-8=-24. 

Again,  to  get  "3  from  1,  this  positive  unit  must  be  increased 
3-fold  and  then  have  its  quality  sign  reversed;  'therefore,  to 
multiply  any  number  by  ~3,  first  increase  that  number  3-fold  and 
then  reverse  the  quality  sign. 

E.g.,  since  -3  ="(1  +  1  +  1), 

therefore  8  •  "3  =  "(8  +  8  +  8)  =  ^24 ; 

similarly,  "8  •  "3  means  that  ~8  is  to  be  increased  3-fold  and  then 
have  its  quality  sign  reversed,  but  ~8  increased  3-fold  is  ~24, 
therefore  -g  .  -3  _+24 

From  what  has  just  been  said,  "8  •  3  =-(8  •  3),  8  •  "3  ="(8  •  3), 
and  ~8- -3=+(8-3);  by  the  same  reasoning  as  that  employed 


-18]  POSITIVE  AND  NEGATIVE  NUMBERS  27 

in  these  particular  cases,  it  follows  that,  whatever  the  numbers 
represented  by  a  and  b, 

+a'+b  =  +(a  .  b), 

-a-'^b  =  ~(a  •  b), 

+a  '-b  =  -(a  -b), 

and  ~a  .  "6  =  +(a  •  b). 

These  results  may  be  formulated  in  words  thus :  the  absolute 
value  of  the  product  of  any  two  numbers  is  equal  to  the 
product  of  their  absolute  values,  and  this  product  is  posi- 
tive if  the  factoids  have  like  quality  signs,  otherwise  it  is 
negative. 

Note  1.  Since  a  succession  of  multiplications*  is  to  be  performed  by  first 
getting  the  product  of  the  first  two  numbers,  then  multiplying  this  product  by 
the  next  number,  and  so  on  (cf.  §  8),  tlierefore,  by  the  successive  application  of 
the  principle  established  for  the  product  of  two  numbers,  it  follows  that  the  abso- 
lute value  of  a  continued  product  is  the  product  of  the  absolute  values  of  the 
factors,  and  this  product  is  negative  if  it  contains  an  odd  number  of  negative 
factors,  otherwise  it  is  positive. 

E.fj.,  5  •  -3  •  -2  .  7  =  -15  •  -2  •  7  =  30  .  7  =  210=  +(5  .3.2.  7). 

Note  2.  From  Note  1  it  follows  that  odd  powers  {i.e.,  powers  whose  expo- 
nents are  odd  numbers)  of  negative  numbers  are  negative,  while  even  powers  of 
negative  numbers  are  positive,  and  all  powers  of  positive  numbers  are  positive. 

E.g.,  (-2)2  =  +4,  (-2)3  =  -8,  (-2)*  =  +16,  etc. 


EXERCISES 

Find  the  value  of  each  of  the  following  indicated  products : 

1.  5-3.  5.   -7f--6.  9.   -2c.  3c. 

2.  -6  .  4.  6.   -m  '  -5.  10.   "3  •  4  •  -6  •    2. 

3.  -7.-2.  7.   -4  a- 3.  11.   3.-A;.-x-4a. 

4.  12  •    9.  8.    -12  .  -3  X.  12.    (-3)2 .5.-2. 

13.  In  the  above  products,  how  does  the  absolute  value  of  the  product 
compare  with  the  product  of  the  absolute  values  of  the  factors?  What 
is  meant  by  the  absolute  value  of  a  number? 


*  A  succession  of  multiplications  such  as  3  •  5  •  9  •  4  •••  is  often  called  a  con- 
tinued product. 


28  ELEMENTARY  ALGEBRA  [Ch.  II 

14.  If  two  numbers  have  like  signs  (both  plus,  or  both  minus),  what 
is  the  sign  of  their  product?  If  they  have  unlike  signs,  what  is  the  sign 
of  their  product  ? 

15.  In  the  continued  product  of  Ex.  10  above,  what  is  the  sign  of  the 
product  of  the  first  two  factors?  of  this  product  multiplied  by  the  next 
factor  ?   of  this  product  by  the  next  factor  ? 

16.  Can  the  sign  of  a  continued  product  be  ascertained  without  actu- 
ally performing  the  multiplication?  How?  What  is  the  sign  of  the 
result  in  Ex.  10  above?  in  Ex.  11?  in  Ex.  12?  If  a  continued  product 
has  five  negative  factors,  what  is  the  sign  of  the  result? 

17.  Define  the  product  of  two  numbers,  and  on  the  basis  of  your 
definition  prove  that  the  sign  of  the  product  -4-7  is  negative.  Also 
that  the  sign  of  the  product  -4  •  "7  is  positive. 

18.  How  is  -5  obtained  from  the  positive  unit?  How  then  is  the 
product  8  •  -5  obtained  ?  the  product  "8  •  -5  ?  Show  that  -2.-2.-2-  -2, 
i.e.,  (-2)4,  is  16;  also  that  (-2)5  = -32.  What  is  the  sign  of  ("6)8? 
of  (-2)4.  (-3)2?   of  (-1)10? 

19.  Define  a  continued  product,  and  state  the  order  in  which  its 
multiplications  are  to  be  performed.  What  is  an  odd  power  of  a 
number  (cf .  §  7)  ?   an  even  power  ? 

Find  the  value  of  (a  +  b)  -  (x  —  y)  : 

20.  When  a  =  2,  &  =  -3,  x  =  -4,  and  y  =  6. 

21.  When  a  =  |,  b  =  -2a,  x  =  -6,  and  y  =  -10. 

22.  When  a  =  -4,  &  =  6,  x  =  ah,  and  y  =  "12. 

23.  When  a  = -4,  b  =  a'^,  x=  Sa,  and  y  =  2a\ 

19.  Division  of  algebraic  numbers.  Since  division  is  the  inverse 
of  multiplication  [cf.  §  3  (iv)],  therefore  the  results  of  §  18  may 
be  used  to  show  how  to  divide  algebraic  numbers. 

For  example,  to  divide  +24  by  ~3  means  to  find  the  number 
which  being  multiplied  by  ~3  will  produce  +24 ;  but,  by  §  18, 
this  number  is  "8 ;  hence 

+24  -f-  -3  =  -8. 
And,  in  general,  whatever  the  numbers  represented  by  a  and  h, 


+(a 

.6)- 

-  +6  =  +a. 

+(« 

.6)- 

--h  =  ~a, 

-(« 

.6)- 

-+h  =  -a, 

and  -(a 

.6)- 

--h  =  +a. 

18-20]  POSITIVE  AND  NEGATIVE  NUMBERS  29 

These  results  may  be  formulated  in  words  thus :  the  absolute 
value  of  the  quotient  of  two  numhers  is  the  quotient  of 
their  absolute  values,  and  this  quotient  is  positive  if  the 
dividend  and  divisor  have  like  signs,  otherwise  it  is 
negative. 

EXERCISES 

Find  the  value  of  each  of  the  following  indicated  quotients : 

1.  14 -r- 2.  4.  -31 --If.  7.   15-i--l. 

2.  14  H-  -2.  5.   -24  -  9.  8.   -365  -  -9^. 

3.  -18 -=-4^.  6.    (-6)2 -(-2)3.  9.   "63  a^  - -7. 

10.  Of  what  operation  is  division  the  inverse?  What  is  an  inverse 
operation  ?  In  an  exercise  in  division,  what  is  it  that  corresponds  to  the 
product  in  multiplication  ?  How  may  the  correctness  of  an  exercise  in 
division  be  tested  ? 

11.  If  the  dividend  is  positive,  and  the  divisor  negative,  what  is  the 
sign  of  the  quotient?  If  the  dividend  is  positive,  how  do  the  signs  of 
divisor  and  quotient  compare  ?  if  the  dividend  is  negative  ? 

Find  the  value  of  each  of  the  following  expressions : 

12.  24  -  28  ^  -7  +  -16  -^  -4  --3.        13.  -8  •  -6  ->■  24  -  27  -^  -6  ^  3. 

14.  {28  -f-  -7  -  2  .  (-4  -  2)  +  24}  -j-  (-2)3. 

Verify  that  «±^  .  ^Ln^  ^  ^^ : 
x-\-y    x-y      x^-y^ 

15.  When  a  =  6,  6  =  2,  x  =  10,  and  ?/  =  6. 

16.  When  a  =  -8,  6  =  12,  a;  =  -9,  and  y  =  'l. 

20.  Small  quality  signs  (+  and  -)  dispensed  with.  To  distinguish 
sharply  between  the  positive  and  the  negative  quality  of  numbers, 
and  at  the  same  time  to  avoid  confusing  signs  of  quality  with  the 
signs  of  the  operations  of  addition  and  subtraction,  the  small  plus 
and  minus  signs  (+  and  ~)  have  thus  far  been  employed. 

In  order  to  simplify  this  notation,  which  is  manifestly  some- 
what cumbersome,  the  larger  plus  and  minus  signs  (+  and  — )  may 
in  future  be  employed  to  indicate  both  the  quality  of  numbers, 
and  also  the  operations  of  addition  and  subtraction.  A  number 
without  a  quality  sign  attached  to  it  will  continue  to  mean  a 


30  ELEMENTARY  ALGEBRA  [Ch.  II 

positive  number,  while  a  negative  number  will  be  indicated  by 
writing  the  minus  sign  before  the  numeral,  and  inclosing  both 
the  numeral  and  its  sign  in  a  parenthesis  when  the  parenthesis  is 
necessary  to  avoid  ambiguity :  the  quality  sign  —  is  never  omitted. 

With  this  simpler  notation :  5  means  the  same  as  +5 ;  a  the  same  as  +a ;  —  8, 
or  (—8),  the  same  as  -8;  9  —  5—  (—3)*  the  same  as  +9  — +5  — -3,  etc. 

In  general  it  may  be  said  that  the  sign  prefixed  to  a  number  indicates  an  opera- 
tion unless  that  number  stands  alone,  or  stands  first  among  several  which  are  to 
be  united,  or  is  inclosed,  together  with  its  sign,  in  a  parenthesis. 

EXERCISES 

1.  In  the  expression  +5  +  '*"3  —  +4,  which  are  signs  of  quality  and 
which  are  signs  of  operation? 

.  2.  Rewrite  the  expression  in  Ex.  1,  omitting  the  quality  signs.  Has 
this  change  in  the  writing  really  made  any  change  in  the  quality  of  the 
numbers  ? 

3.  Answer  questions  1  and  2  with  regard  to  the  expression  +5  —  +3  +  +4. 

4.  Could  all  the  quality  signs  in  the  expression  +15  —  +3  +  -8  be 
omitted  without  changing  the  meaning  of  the  expression?  Which  of 
these  signs  might  be  omitted?  When  no  quality  sign  is  written,  what  is 
the  quality  of  the  number? 

5.  If  the  expression  in  Ex.  4  be  written  so  as  to  use  only  the  larger 
signs,  is  a  parenthesis  necessary  to  preserve  the  meaning?  Write  the 
expression  so.  Also  answer  the  same  questions  with  regard  to  the 
expression  a;  —  "5  +  ~8. 

6.  Show  that  the  expression  a:  —  "5  +  -8  is  equal  to  ar  +  5  —  8,  wherein 
both  5  and  8  are  positive  numbers,  and  the  signs  +  and  —  indicate 
operations. 

21.  Algebraic  expressions.  Terms.  In  the  course  of  operations 
with  algebraic  numbers,  it  often  happens  that  the  expression  for 
a  number  does  not  consist  of  a  single  symbol,  but  rather  of  a 
combination  of  such  symbols. 

E.g.,  if  a  and  h  represent  numbers,  then  ab,  a  +  b,  and  a^  —  3  aft^  also  represent 
numbers. 

*  By  §§  16  and  17  this  expression  equals  9  —  5  +  3,  which  is  7.  In  this  connec- 
tion attention  may  also  be  called  to  the  fact  that  since  a-\-  (  —  b)  =  a  —  b  (§16), 
therefore  such  an  expression  as  a  —  6  may  be  understood  as  meaning  either  that  b 
is  subtracted  from  a,  or  that  —  6  is  added  to  a. 


20-22]  POSITIVE  AND  NEGATIVE  NUMBERS  31 

Such  expressions  for  numbers  as 
a  +  b,   3xy,   m'  +  27i'-5x,   dax" +  —  -10^^ +Saxy',  etc., 

are  called  algebraic  expressions.* 

The  parts  of  an  algebraic  expression  which  are  connected  by 
the  signs  +  and  —  (or,  rather,  these  parts  together  with  the  signs 
preceding  them)  are  called  the  terms  of  the  expression.  Terms 
preceded  by  the  plus  sign  are  called  positive  terms,  while  those 
preceded  by  the  minus  sign  are  called  negative  terms. 

E.g.,  in  the  expression  5cfi-\-Zh  —  lQc^x'^,  there  are  three  terms,  viz.:  5a2, 
+  36,  and  — 10  c^x^ ;  the  first  two  are  positive,  and  the  third  is  negative. 

EXERCISES 

1.  How  many  terms  are  there  in  the  expression 

5  a%  +  2  axif  -  7  mx^  -  26  ? 
What  are  they  ?     Which  are  positive  V     Which  negative  ? 

2.  Answer  the  same  questions  as  in  Ex.  1  with  regard  to  the  expression 

-12  +  7  rrfix^  -  5  a?/^  -  3  a;2  _  §  ahiA 

3.  The  sum  of  two  times  a  number  and  three  times  the  same  num- 
ber is  how  many  times  that  number?  Unite  the  two  terms  3a:  +  5a: 
into  one.  What  single  term  is  equal  to  \  x  —  \  x'l  Is5a:+13a:  —  9x 
equal  to  (5  +  13  -  9)  a:  ?     Why  ? 

22.  Recapitulation.  In  this  chapter  it  has  been  shown  that,  in 
order  to  express  in  a  simple  way  the  relations  between  assets  and 
liabilities,  latitude  north  and  latitude  south  of  the  equator,  tem- 
perature above  zero  and  temperature  below  zero,  in  fact,  between 
any  of  the  things  which  bear  a  relation  of  opposition  to  each  other, 
and  which  are  everywhere  met  with  in  one's  daily  intercourse, 
it  is  advantageous  to  extend  the  number  system  so  as  to  make 
subtraction  always  possible. 

Further  considerations  have  shown  that  the  numbers  needed  to 
make  subtraction  always  possible  are  the  so-called  negative  num- 
bers, and  in  §§  15-19  it  has  been  shown  how  to  interpret  these 
numbers,  and  also  how  to  operate  with  and  upon  them.  A  rapid 
re-reading  of  these  paragraphs  is  recommended. 

*  An  algebraic  expression  is  spoken  of  as  an  expression,  or  as  a  wwrrtfter  accord- 
ing as  the  thought  is  of  the  combined  symbol,  or  of  the  numerical  value  which 
that  symbol  represents. 


CHAPTER   III 
THE   EQUATION 

23.  Definitions.  Although  a  discussion  of  the  fundamental 
principles  relating  to  equations  must  be  postponed  until  more 
of  the  theory  connected  with  algebraic  expressions  has  been 
developed  (see  Chapter  X),  yet  the  importance  of  the  equation 
as  an  instrument  of  investigation  demands  that  it  be  presented  as 
early  as  possible. 

An  equation  has  already  been  defined  [§  3  (i)]  as  a  statement 
that  each  of  two  expressions  has  the  same  value  as  the  other,  i.e., 
it  is  a  statement  that  each  of  these  expressions  represents  the 
same  number.  These  two  expressions  are  called  the  members  of 
the  equation,  and  that  expression  which  is  written  at  the  left 
of  the  sign  of  equality  is  known  as  the  first  member,  while  the 
other  is  known  as  the  second  member. 

E.g.,  8  «  —  21  =  3  a;  +  4  is  an  equation  of  which  8  a;  —  21is  the  first  member,  and 
3  a;  +  4,  the  second  member. 

Manifestly  the  two  members  of  the  equation  just  written  do  not 
represent  equal  numbers  for  all  values  that  may  be  assigned  to 
the  unknown  number  represented  by  x :  indeed  there  is  only  one 
value  of  X  for  which  they  are  equal ;  viz.,  for  a?  =  5.  Hence  such 
an  equation  is  called  a  conditional  equation;  it  is  an  equation  only 
on  condition  that  a;  =  5. 

An  equation  which  is  true  for  all  values  that  may  be  assigned  to 
its  letters  is  called  an  identical  equation  or,  more  briefly,  an  identity. 
To  indicate  that  an  equation  is  an  identity,  rather  than  a  condi- 
tional equation,  the  sign  =  may  be  used  instead  of  =  to  connect 
the  two  members. 

E.g.,  3a;  +  5  —  x  =  2x4-7  —  2  and  ax"^  +  &  —  ax"^  =  b  are  identities.  Many  other 
examples  of  identities  will  present  themselves  in  the  following  pages. 

82 


23-24]  THE  EQUATION  33 

The  process  of  deducing  from  any  conditional  equation  the 
values  that  must  be  substituted  for  the  unknown  number  to  make 
the  two  members  equal,  is  called  solving  the  equation,  and  these 
values  themselves  are  called  the  solutions  or  roots  of  the  equation. 

Note.  The  final  test  as  to  whether  a  number  is  or  is  not  a  root  of  a  given 
equation  is  to  substitute  that  number  for  the  letter  representing  the  unknown 
number  in  the  equation  ;  if  this  substitution  satisfies  the  equation,  i.e.,  if  it  makes 
the  two  members  reduce  to  the  same  number,  then  it  is  a  root,  otherwise  it  is  not. 
E.g.,  5  is  a  root  of  the  equation  8a;  —  21  =  3x+4,  because  substituting  5  for  x 
satisfies  this  equation. 

24.  Some  axioms  and  their  use.  The  following  principles, 
usually  called  axioms,  are  useful  in  solving  equations. 

(1)  If  equals  he  added  to  or  subtracted  from  equals,  the 
results  will  he  equal.* 

(2)  If  equals  he  multiplied  or  divided  hy  equals,  the 
results  will  he  equal  A 

The  application  of  these  axioms  to  the  solution  of  equations  is 
illustrated  by  the  following  examples :  X 

Ex.  1.  If  8  a;  —  21  =  3  X  +  4,  find  the  value  of  x ;  i.e.,  solve  this  equation. 

Solution 

Since  8a:-21  =  3a:  +  4, 

therefore  8  x  -  21  +  21  =  3  a;  +  4  +  21,                  [Axiom  (1) 

i.e.,  8  a:  =  3  a:  +  25, 

and  therefore  8a;  —  3a;  =  3a:  +  25  —  3x,               [Axiom  (1) 

i.e.,  5  a:  =  25, 

whence  a:  =  5.                                      [Axiom  (2) 

Verification.  Substituting  5  for  x  in  the  original  equation,  each 
member  reduces  to  19 ;  that  is,  the  substitution  of  5  for  x  satisfies  this 
equation,  and  5  is  therefore  a  root  of  it. 

*  Equal  numbers  are  really  the  same  number  ;  such  numbers  may,  of  course, 
be  expressed  in  different  ways  (e.g.,  19  +  5,  3  •  8,  and  5-5  —  1  each  express  24), 
but  they  are,  nevertheless,  the  same  number,  and  the  self-evidence  of  these 
axioms  rests  upon  that  fact. 

t  It  is  not  permissible,  however,  to  divide  by  zero. 

J  See  footnote,  p.  6. 


34  ELEMENTARY  ALGEBRA  [Ch.  Ill 

Ex.  2.    Solve  the  equation  ^ x  +  12  -\-  7  x  =  \x  -  l()\  -  4:X. 

Solution 

Since  |a:+12  +  7x  =  |x-10i-4a:, 

therefore,  multiplying  each  member  by  6, 

,  4  a;  +  72  +  42  a:  =  3  a;  -  62  -  24  X,  |;  Axiom  (2) 

i.e.,  46  a:  +  72  =  -  21  a;  -  62, 

and  therefore,  subtracting  72  from  each  member, 

46  a;  =  -  21  x  -  62  -  72  [Axiom  (1) 

zz  -  21  a;  -  134, 

and,  adding  21  x  to  each  member, 

67  a;  =  -  21  a:  -  134  +  21  a:  =  -  134, 

whence  a:  =  —  2.  [Axiom  (2) 

Verification.  Since  the  substitution  of  —  2  for  x  satisfies  the  origi- 
nal equation,  therefore  —  2  is  a  root  of  that  equation. 

EXERCISES 

3.  Define  an  equation.  Also  distinguish  between  a  conditional 
equation  and  an  identity.  Give  an  illustrative  example  of  each  of 
these  two  kinds  of  equations.  Is  2 ax  +  3a  =  a(4ar  +  3)—  2 aa;  a  con- 
ditional equation  or  an  identity? 

4.  What  are  the  members  of  an  equation?  Which  is  called  the  first 
member?  What  is  the  other  member  called?  What  is  meant  by  a  root 
of  an  equation?     Illustrate  your  answers  by  suitable  examples. 

5.  What  is  meant  by  solving  an  equation?  Describe  briefly  the 
process  of  solving  an  equation.  State  the  axioms  which  have  thus  far 
been  employed  in  solving  equations.  Illustrate  your  answers  by  suitable 
examples. 

6.  How  may  the  correctness  of  a  solution  (root)  be  verified  ?  Show 
that  4  is  a  root  of  7  a:  -  10  =  4  x  +  2.  Is  2  a  root  of  a:2-5a:  +  6  =  0? 
Is  3  also  a  root  of  this  last  equation? 

Solve  the  following  equations,  give  the  reasons  for  each  step  of  the 
work,  and  test  the  correctness  of  the  roots : 

7.  3  a;  -F  2  =  a;  +  30.  9.   2  a:  +  -  =  — • 

3       6 

8.  7  a:  -  55  =  18  -  2  a:  -  1.  10.    5  a:  -  3j  a:  =  17  -  a:. 


24-25]  THE  EQUATION  35 

11.  If  the  second  member  of  an  equation  be  multiplied  by  any  num- 
ber, say  4,  what  must  be  done  to  the  first  member  in  order  to  preserve 
the  equality?  If  any  given  number  be  added  to  either  member,  what 
must  be  done  to  the  other  member?    Why? 

12.  If  2  a  be  subtracted  from  each  member  of  the  equation  5  a:  +  2  a 
=  3  a;  +  4  &,  what  is  the  resulting  equation  ?  What  does  this  show  with 
reference  to  removing  a  term  from  the  first  to  the  second  member  of  an 
equation?  Is  the  same  thing  true  when  a  term  is  removed  from  the 
second  member  to  the  first?  Show  this  by  adding  -3  x  to  each  member 
of  the  given  equation. 

25.  Transposition ;  directions  for  solving  equations.  Eemoving  a 
term  from  one  member  of  an  equation  to  the  other  is  spoken  of 
as  transposing  that  term.  It  has  doubtless  been  observed,  in  the 
solutions  of  the  equations  of  §  24,  that  a  term  may  be  trans- 
posed froin  one  member  of  an  equation  to  the  other  by 
merely  reversing  its  sign. 

This  fact  may  be  formally  proved  as  follows :  let  any  term  of  either  member 
{e.g.,  the  first)  of  any  given  equation  be  represented  by  k,  —  this  term  may  be 
positive  or  negative,  and  may  contain  any  number  of  letters,  —  and  let  the  remain- 
ing terms  of  the  first  member  of  this  equation  be  represented  hy  M,  and  its  second 
member  by  N ;  then  the  equation  is 

M-\-k  =  N. 
Subtracting  k  from  each  member  of  this  equation,  it  becomes,  by  axiom  (1), 

M=N-k, 
i.e.,  the  term  k  has  disappeared  from  the  first  member  of  the  given  equation,  but 
has  reappeared,  with  its  sign  reversed,  in  the  second  member. 

The  following  simple  directions  may  now  be  given  for  solving 
such  equations  as  those  considered  in  §  24. 

(1)  If  the  equation  contains  fractions,  multiply  both  of 
its  mcTnbers  by  the  legist  common  multiple  of  the  denomina- 
tors of  these  fractions  (axiom  2);  this  is  usually  spoken  of  as 
clearing  the  equation  of  fractions. 

(2)  Transpose  all  the  terms  containing  the  unknown 
number  to  the  first  member  of  the  equation,  and  all  other 
terms  to  the  second  member. 

(3)  Unite  the  terms  of  each  member,  and  then  divide 
both  members  by  the  coefficient '^  of  the  unknown  number. 

*  The  coefficient  of  the  unknown  number  is  the  factor  which  multiplies  it. 


36  ELEMENTARY  ALGEBRA  [Ch.  Ill 

(4)  Substitute  the  value  thus  found  for  the  unhnown 
number  in  the  given  equation;  if  this  satisfies  the  equa- 
tion, then  it  is  a  root  of  the  equation,  otherwise  it  is  not. 

These  directions  may  be  illustrated  by  solving  again  Ex.  2  of 
§  24,  thus : 

Given  lx-^12-^1  x=lx-l0}-4:x\^ 

multiplying  the  given  equation  by  6  to  clear  it  of  fractions,  it  becomes 
4  a:  +  72  +  42  a:  =  3  a;  -  62  -  24  X,    [Axiom  (2) 
whence,  transposing,  4  x+42  x—^x  +  24  x  =  —  62  —  72, 
i.e.,  67  a;  =  — 134 ;  [Uniting  terms 

therefore,  dividing  by  67,  x  =  —  2 ; 

and  this  value  of  x  proves,  on  substitution,  to  be  a  root  of  the  given 
equation. 

EXERCISES 

Solve  the  following  conditional  equations,  and  verify  the  results  : 

1.  12a:+5x  +  20-8a:  =  48  +  3a:-4.  5.    §^  +  5  =  91  -  lOar. 

2.  3(x-5)*+4a;  +  8  =  5(4ar-20).  ^    7^+2-^=17 

3.  5(2a:-10)+7ar-15  =  20a:.  „  o„ 

^  7.   8  +  2v  +  '^=l|  +  ^- 

3  7  '  Q.   ^k-lQ=.2k-\-ll. 

9.   Uk-20  +  lk-2  =  Qk  +  V- 

10.   2t;+^-^  +  14  =  7.-^  +  ^-i^. 
2       4  4      7        2 

26.  Problems  leading  to  equations.  A  problem  is  a  question  pro- 
posed for  solution;  it  always  asks  to  find  one  or  more  numbers 
which  at  the  beginning  are  unknown,  and  it  states  certain  relations 
(conditions)  between  these  numbers,  by  means  of  which  their 
values  may  be  determined. 

The  process  of  solving  problems  has  already  been  illustrated 
in  §  6,  — which  should  now  be  re-read.     The  important  steps  are : 

(1)  Represent  one  of  the  unhnown  numbers  involved  in 
the  problein  by  some  letter,  as  x. 

(2)  From  the  verbal  conditions  of  the  problem  find  alge- 
braic expressions  for  the  other  unhnown  numbers,  and 
form  two  such  expressions  that  are  equal  to  each  otlwr. 

*  That  3(a;  —  5)  =  3  a;  — 15  may  for  the  present  be  assumed ;  it  is  proved  in  §  39. 


25-26]  PROBLEMS  37 

(3)  With  these  two  equal  expressions,  form  an  equation,  ~ 
called  the  equation  of  the  problem. 

(4)  Solve  this  equation  and  verify  the  correctness  of  the 
result. 

These  steps  are  illustrated  in  the  solutions  of  the  following 
problems : 

Prob.  1.  The  sum  of  the  ages  of  a  father  and  son  is  54  years,  and  the 
father  is  24  years  older  than  the  son.     How  old  is  each? 

Solution 
The  conditions  of  this  problem,  stated  in  verbal  language,  are : 

(1)  The  number  of  years  in  the  father's  age  plus  the  number  of  years 
in  the  son's  age  is  54.  / 

(2)  The  number  of  years  in  the  son's  age  plus  24  equals  the  number 
of  years  in  the  father's  age. 

To  translate  these  conditions  into  symbolic  language,  let  x  represent 
the  number  of  years  in  the  son's  age,*  then  by  the  second  condition  the 
number  of  years  in  the  father's  age  is  a:  +  24,  and  by  the  first  condition 

a;  +  24  +  a;  =  54, 

which  is  the  equation  of  the  problem. 

From  this  equation  it  is  found  that  x  =  15,  which  is  the  number  of 
years  in  the  son's  age,  and  a:  +  24  =  39,  the  number  of  years  in  the 
father's  age.  By  substituting  these  numbers  it  is  found  that  they  satisfy 
the  two  given  conditions  of  the  problem  and  are,  therefore,  its  solution. 

Note.  It  maybe  worth  remarking  that  it  was  not  necessary,  but  only  con- 
venient, to  let  z  stand  for  the  number  of  years  in  the  son's  age. 

Thus,  if  X  represents  the  number  of  years  in  the  father's  instead  of  in  the  son's 
age,  then  the  given  conditions  translated  into  algebraic  language  become  : 

(1)  54  —  cc  =  the  number  of  years  in  the  son's  age,  and 

(2)  54  — a; +  24  =  X', 

which  is  the  equation  of  the  problem. 

From  this  equation  it  is  found  that  x  =  39,  whence  54  —  x  =  15 ;  these  are  the 
same  numbers  as  obtained  before. 

Again,  if  3x  were  chosen  to  represent  the  number  of  years  in  the  son's  age, 
then  the  equation  of  the  problem  would  be 

3  X  +  24  -h  3  X  =  54, 
whence  x  =  5  and  3  x  =  15,  the  son's  age,  and  3  x  +  24  =  39,  the  father's  age. 

*  It  is  to  be  carefully  noted  that  x  represents  a  number ;  it  does  not  represent 
the  son's  age,  but  represents  the  number  of  years  in  the  son's  age. 


38  ELEMENTARY  ALGEBRA  [Ch.  Ill 

Prob.  2.  A  boy  was  given  39  cents  with  which  to  purchase  3-ceiit 
and  5-cent  postage  stamps,  and  was  told  to  purchase  5  ixiore  of  the  former 
than  of  the  latter.     How  many  of  each  kind  should  he  purchase  ? 

Solution 
The  conditions  of  this  problem,  stated  in  verbal  language,  are : 

(1)  The  total  expenditure  is  39  cents. 

(2)  There  are  to  be  5  more  3-cent  stamps  than  5-cent  stamps. 

To  translate  these  conditions  into  symbolic  language,  let  x  stand  for 
the  number  of  5-cent  stamps  purchased ;  their  cost  is  then  5  x  cents : 
then,  by  the  second  condition,  the  number  of  3-cent  stamps  is  x  +  5,  and 
their  cost  is  (3a:-j-15)  cents;  hence,  by  the  first  condition, 

5  a;  -1-  3  a;  4-  15  =  39, 

which  is  the  equatjon  of  this  problem. 

The  solution  of  this  equation  gives  a;  =  3,  the  number  of  5-cent  stamps, 
and  X  -{-  D  =  S,  the  number  of  3-cent  stamps ;  and  it  is  easily  verified  by 
substitution  that  these  two  numbers  do,  in  fact,  satisfy  both  the  condi- 
tions of  the  problem ;  hence  they  are  the  numbers  sought. 

Prob.  3.  If  a  certain  number  be  diminished  by  6,  and  2  times  this 
difference  be  added  to  5  times  the  number,  the  result  will  equal  88  minus 
3  times  the  number.     What  is  the  number  ? 

Solution 

To  form  the  equation  of  this  problem,  let  x  represent  the  given  number ; 
then  5  times  the  number  is  5  x,  the  number  diminished  by  6  is  a:  — 6,  etc., 
and  the  given  condition  becomes 

5  ar  -}-  2(a:  -  6)   =  88  -  3  a:, 

whence  5  a;  -|-  2  a;  -  12  =  88  -  3  a:, 

and,  transposing,  5a:  +  2a:-F3a:=88  4-12,  » 

i.e.,  10  a:  =  100, 

and,  therefore,  x  =  10, 

which,  on  verification,  proves  to  be  the  required  number. 

Prob.  4.  A  number  consists  of  two  digits  whose  sum  is  5 ;  if  the  digits 
be  interchanged,  the  number  will  be  diminished  by  9.  What  is  the 
number  ? 

Solution 

To  form  the  equation  of  this  problem,  let  x  represent  the  digit  in 
units'  place ;  then,  by  the  first  condition,  5  —  a;  will  represent  the  digit  in 


26]  PROBLEMS  39 

tens'  place ;  therefore,  the  number  is  10(5  —  x)  +  x,  —  compare  Ex.  6,  §  5, — 
and  the  number  formed  by  interchanging  the  digits  is  10  a;  +  (5  —  x). 
The  second  condition  then  gives 

10  X  +  (5  -  x)  =  10(5  -  x)  +  a:  -  9, 
whence  x  =  2,    the  digit  in  units'  place, 

and  5  —  X  =  3,    the  digit  in  tens'  place. 

These  two  digits  are  found  to  satisfy  both  the  conditions  of  the  prob- 
lem, hence  the  number  sought  is  32. 

PROBLEMS 

5.  Divide  28  into  two  parts  whose  difference  is  4. 

6.  The  sum  of  two  numbers  is  63,  and  the  larger  exceeds  the  smaller 
by  17.     What  are  the  numbers  ? 

7.  If  I  of  a  certain  number  exceeds  I-  of  that  number  by  8,  what  is 
the  number  ? 

8.  Divide  48  into  two  parts  such  that  twice  the  larger  part  equals  5 
times  the  smaller  part. 

9.  A  man  who  is  32  years  old  has  a  son  who  is  8  years  old ;  how 
many  years  hence  will  the  father  be  3  times  as  old  as  his  son  ? 

10.  On  being  asked  his  age,  a  gentleman  replies  that  liis  age  5  years 
hence  will  be  twice  as  great  as  it  was  20  years  ago ;  how  old  is  he? 

11.  How  old  is  a  person  if  20  years  hence  his  age  will  be  less  by  5 
years  than  twice  his  present  age  ? 

12.  If  16  be  added  to  a  certain  number,  the  result  will  be  the  same  as 
it  would  be  if  7  times  the  number  were  subtracted  from  56 ;  what  is  the 
number  ? 

13.  If  6  times  a  certain  number  is  as  much  less  than  62  as  3  times  this 
number  exceeds  19,  what  is  the  number? 

14.  Of  four  given  numbers  each  exceeds  the  next  below  it  by  3,  and 
the  sum  of  these  numbers  is  58 ;  find  the  numbers. 

15.  Mary  is  25  years  younger  than  her  mother,  but  if  she  were  one 
year  older  than  she  is  she  would  be  i  as  old  as  her  mother ;  what  is  the 
age  of  each  ? 

16.  The  sum  of  three  numbers  is  25;  the  first  of  these  numbers  is 
gi-eater  by  5  than  the  third,  but  only  -^  as  great  as  the  second ;  find  the 
numbers. 

17.  Divide  f  2200  among  A,  B,  and  C  in  such  a  way  that  B  shall  have 
twice  as  much  as  A,  and  C  $200  more  than  B. 


40  ELEMENTARY  ALGEBRA  [Cii.  Ill 

18.  Divide  $351  among  three  persons  in  such  a  way  that  for  every  dime 
the  first  receives,  the  second  shall  receive  25  cents,  and  the  third  a  dollar. 

19.  Three  boys  together  have  140  marbles ;  if  the  second  has  twice  as 
many  as  the  first,  but  only  half  as  many  as  the  third,  how  many  marbles 
has  each  boy  ? 

20.  After  taking  3  times  a  certain  number  from  11  times  that  number, 
and  then  adding  12  to  the  remainder,  the  result  is  less  than  117  by  7  times 
the  number ;  what  is  the  number  ? 

21.  A  number  consists  of  two  digits  whose  sum  is  8,  and  if  36  be  sub- 
tracted from  this  number  the  order  of  its  digits  will  be  reversed  ;  what 
is  the  number? 

22.  In  a  certain  two-digit  number  the  tens'  digit  is  twice  the  units' 
digit,  and  the  number  formed  by  interchanging  the  digits  equals  the 
given  number  diminished  by  18 ;  w^hat  is  the  number  ? 

23.  In  a  three-digit  number  the  tens'  digit  exceeds  the  hundreds' 
digit  by  3,  the  units'  digit  is  4  less  than  twice  the  hundreds'  digit,  and 
interchanging  the  units'  and  tens'  digits  decreases  the  number  by  45 ; 
what  is  the  number  ? 

24.  A  two-digit  number  is  equal  to  7  times  the  sum  of  its  digits,  and 
the  tens'  digit  exceeds  the  units'  digit  by  3 ;  what  is  the  number? 

25.  A  merchant  owes  A  three  times  as  much  as  he  owes  B,  he  owes  C 
twice  as  much  as  he  owes  A,  and  he  owes  D  as  much  as  he  owes  A  and  B 
together ;  if  the  sum  of  his  indebtedness  to  A,  B,  C,  and  D  is  $28,000, 
how  much  does  he  owe  each? 

26.  Two  clerks,  A  and  B,  have  the  same  salary ;  A  saves  i  of  his,  but 
B,  by  spending  $150  more  than  A  each  year,  saves  only  $350  in  7  years ; 
what  is  the  salary  of  each? 

27.  A  merchant  bought  some  eggs  at  the  rate  of  2  for  3  cents,  he  then 
bought  J  as  many  more  at  the  rate  of  6  for  5  cents,  and  later  sold  them 
all  at  the  rate  of  3  for  4  cents,  thereby  losing  6  cents ;  how  many  did  he 
buy? 

28.  If  I  of  a  number  is  as  much  less  than  the  number  itself  as  |  of 
the  number  is  less  than  65,  what  is  the  number  ? 

29.  The  sum  of  three  consecutive  integers  is  51 ;  what  are  these  thiee 
numbers  (cf.  Ex.  8,  §  5)?  Show  that  the  sum  of  any  three  consecutive 
integers  is  3  times  the  second  of  these  integers. 

30.  The  sum  of  four  consecutive  odd  integers  is  80;  what  are  these 
four  numbers?  Prove  that  the  sum  of  any  four  odd  integers  is  an  even 
integer. 


26]  PROBLEMS  41 

31.  M  can  do  a  certain  piece  of  work  in  8  days,  and  N  can  do  it  in 
12  days;  iu  how  many  days  can  both  do  it  when  working  together 
[cf.  §9(ii)]? 

32.  If  M  begins  the  work  mentioned  in  Prob.  31,  and,  after  working  a 
certain  number  of  days  at  it,  turns  it  over  to  N  to  finish,  and  the  entire 
piece  of  work  is  done  in  10  days,  how  long  did  each  work  at  it  ? 

33.  A  country  club  consisting  of  200  members,  having  decided  to 
build  a  new  club  house,  assessed  each  of  its  members  a  certain  sum  for 
that  purpose ;  meanwhile  the  membership  was  increased  by  50,  and  it 
was  then  found  that  the  assessment  could  be  reduced  by  $10;  what  was 
the  cost  of  the  proposed  house? 

34.  A  real  estate  dealer  purchased  three  houses,  paying  1|  times  as 
much  for  the  second  as  for  the  first,  and  If  times  as  much  for  the  third  as 
for  the  first ;  if  the  difference  between  the  cost  of  the  second  and  third 
was  11500,  what  was  the  cost  of  each? 

35.  A  gentleman  left  his  property,  valued  at  $800,000,  to  be  divided 
among  three  colleges;  if  the  first  was  to  receive  $30,000  more  than  the 
second,  and  the  third  half  as  much  as  the  other  two  together,  how  much 
was  each  to  receive? 

36.  Five  boys  had  agreed  to  purchase  a  pleasure-boat,  but  one  of  them 
withdrew,  and  it  was  then  found  that  each  of  the  remaining  boys  had  to 
pay  $2  more  than  would  have  been  necessary  under  the  original  plan; 
how  much  did  the  boat  cost? 

37.  A  lady  having  already  spent  $10  more  than  |  of  her  money  made 
further  purchases  amounting  to  $10  more  than  f  of  what  then  remained, 
and  found  that  she  had  only  $2  left;  how  much  had  she  at  first? 

38.  A  laborer  was  engaged  to  do  a  certain  piece  of  work  on  condition 
that  he  was  to  receive  $2  for  every  day  that  he  worked,  and  to  forfeit 
50  cents  for  every  day  that  he  was  idle ;  at  the  end  of  18  days  he  received 
$28.50.     How  many  days  did  he  work? 

39.  A  certain  number  being  subtracted  from  50,  and  also  from  84,  it 
is  found  that  f  of  the  first  of  these  remainders  exceeds  |  of  the  second  by 
47 ;  what  is  the  number  ? 


CHAPTER   IV 

ADDITION  AND  SUBTRACTION  OF  ALGEBRAIC  EXPRES- 
SIONS —  PARENTHESES 

I.    ADDITION 

27.  Monomials,  binomials,  etc. ;  coefficients.  An  algebraic  expres- 
sion consisting  of  but  one  term*  is  called  a  monomial,  while  one 
consisting  of  two  or  more  terms  is  called  a  polynomial.  A  poly- 
nomial consisting  of  only  two  terms  is  usually  called  a  binomial, 
and  one  consisting  of  three  terms,  a  trinomial ;  but  to  polynomials 
consisting  of  more  than  three  terms  it  is  not  customary  to  give 
special  names  corresponding  to  binomial  and  trinomial. 

E.g.,  2ax^,—7m^p^,  and  Sbx^i/^  are  monomials;  x^  +  Sy,  5  m  — 2  z^,  and 
—  3  a62  —  f  ^Sy4  are  binomials ;  'and  2  a;8  +  4  ay  —  5  62,  2  s*  —  6  ?/  +  3  m^x^,  and 
x  +  'dt  —  j  abx'^  are  trinomials. 

If  a  term  is  composed  of  several  factors,  any  one  of  its  factors, 
or  the  product  of  two  or  more  of  them,  is  called  the  coefficient  of 
the  product  of  the  remaining  factors. 

E.g.,  in  the  term  5  axy^,  the  coefficient  of  axy^  is  5,  the  coefficient  of  xy^  is  5  a, 
the  coefficient  of  5  xy'^  is  a,  etc. 

A  coefficient  consisting  of  Arabic  characters  only  is  a  numerical 
coefficient,  while  one  that  contains  one  or  more  literal  factors  is  a 
literal  coefficient. 

E.g.,  in  the  term  ~3ax^y^,  the  numerical  coefficient  of  ax^y*  is  — 3,  but  —3a 
and  3  ay^  are  literal  coefficients  of  x^y^  and  —  x^  respectively. 

Note.  The  word  "coefficient"  is  usually  understood  to  mean  "numerical 
coefficient,"  and  the  sign  (+  or  — )  written  before  a  term  is  usually  regarded  as 
belonging  to  the  numerical  coefficient.  When  no  numerical  coefficient  is  written, 
the  term  is  understood  to  have  the  coefficient  1. 

*  For  the  definition  of  an  "  algebraic  expression,"  and  of  a  *'  term,"  see  §  21. 

42 


27-28]  ADDITION  43 

28.  Positive  and  negative  terms  ;  like  and  unlike  terms.  A  term 
whose  sign  is  +  is  called  a  positive  term,  and  one  whose  sign  is  — 
is  called  a  negative  term.  If  the  first  term  of  an  algebraic  expres- 
sion is  positive,  its  sign  is  usually  omitted,  but  the  sign  of  a  nega- 
tive term  is  never  omitted. 

Note.  As  has  already  been  pointed  out,  the  letters  in  an  algebraic  expression 
may  represent  any  numbers  whatever,  —  they  may  be  positive  or  negative,  even 
or  odd,  integers  or  fractions,  —  and  therefore  an  algebraic  expression  which  is 
fractional  in  appearance  may  have  an  integral  value,  and  vice  versa ;  so  too  a 
term  which  is  positive  in  appearance  may  still,  for  certain  values  of  the  letters 
involved  in  it,  have  a  negative  value,  and  vice  versa. 

Terms  which  either  do  not  differ  at  all,  or  which  differ  only  in 
their  numerical  coefficients,  or  in  their  quality  signs,  are  called 
like  terms,  and  also  similar  terms;  terms  which  differ  in  other 
respects  are  called  unlike  terms,  and  also  dissimilar  terms. 

E.g.,  Zxhj,  hx^ij,  and  —Ix^y  are  like  terms,  while  2 ax,  —ob^x^y,  and  Sxy^ 
are  unlike  terms. 

Like  terms  must  contain  the  same  letters,  and  these  letters  must  be  affected 
with  the  same  exponents,  but  they  may  differ  in  their  signs  and  also  in  their 
coefficients. 

EXERCISES 

1.  What  is  the  coefficient  of  a^x  in  each  of  the  following  expressions : 

3a%,  -  6a^x,  a%  4  a^&x,   -  4a%,  ^"  "  "^,  and  -9a^x? 

1  m 

2.  Which  of  the  above  coefficients  are  literal  and  which  numerical? 
Which  of  the  terms  in  Ex.  1  are  positive  and  which  negative? 

3.  Do  the  positive  terms  in  Ex.  1  necessarily  represent  positive  num- 
bers for  all  values  that  may  be  assigned  to  the  letters  involved  ?  Try 
a  =  3  and  x  ■=—2. 

4.  What  is  the  coefficient  of  x—yin  each  of  the  following  expressions : 
\^{x  -  y),  —  a(x  —  y),  f  m(x  -  y),  and  (4  -  a^)(x  —  y)  ?  Which  of 
these  coefficients  are  numerical?  Which  literal?  Which  of  these  expres- 
sions are  positive  and  which  negative  ?  Try  various  values  for  the  letters 
and  see  whether  the  negative  expressions  necessarily  represent  negative 
numbers. 

5.  Consult  a  good  dictionary  for  the  derivation  of  the  words  "  mono- 
mial," "binomial,"  "trinomial,"  and  "polynomial."  Write  three  mono- 
mials, three  binomials,  three  trinomials,  and  three  polynomials. 

6.  Distinguish  carefully  between  the  meanings  of  5  in  the  expressions 
5  X  and  x^.     What  name  is  given  to  the  5  in  each  of  these  expressions? 


44  ELEMENTARY  ALGEBRA  [Cn.  IV 

7.  What  are  like  terms?  By  what  other  name  are  they  known?  In 
what  respects  may  they  differ  and  still  be  like  terms  ?  Are  3  x^y,  —2x^i/, 
and  I  a;2^  similar?  Are  4:  ax^  and  —  Gbx^  similar?  Are  these  last  two 
terms  similar  if  4a  and  —Qb  are  regarded  as  their  respective  coefficients? 

8.  Write  three  sets  of  like  terms,  some  terms  being  positive  and  some 
negative,  and  each  set  containing  at  least  four  terms. 

29.  Addition  of  monomials.  That  the  sum  of  several  similar 
monomials  may  be  united  into  a  single  term  has  already  been 
illustrated  in  some  of  the  exercises  and  problems  in  the  preceding 
pages ;  this  subject  will  now  be  considered  in  greater  detail. 

Since  5  times  any  given  number,  plus  2  times  that  number,  is 
7  times  the  given  number,  i.e.,  (5  +  2)  times  the  given  number, 
therefore  o  a  -^2  a  =  (o-i-2)  a  =  7  a,  whatever  the  number  repre- 
sented by  a.     So  too  3  mxhj  +  8  mx-y  =  (3  +  8)  mx^y  =  11  mx^y. 

Observe  that  this  reasoning  applies  to  any  two  similar  monomials  whatever. 

Since  the  sum  of  three  or  more  numbers  is  obtained  by  adding 
the  third  to  the  sum  of  the  first  and  second,  the  fourth  to  the  sum 
of  the  first  three,  etc. ,  therefore,  to  add  any  nuinber  of  similar 
monomials,  add  their  coefficients,  and  to  this  result  annex 
the  common  literal  factors. 

It  is  usually  most  convenient  to  write  the  terms  to  be  added 
under  one  another,  as  in  arithmetic,  thus : 

3x?/2  153a2ma;8  l%ak^8 

8  xy^  74  a2mx8  _  7  ak'^s 


11  xy"^  227  a^mx^  11  ak'^s*' 

If  the  monomials  to  be  added  are  dissimilar,  they  cannot  be 
united  into  a  single  term,  but  their  sum  may  be  indicated  in  the 
usual  way ;  e.g.,  the  sum  of  5  a  and  2  car'  is  5  a  -f  2  ca^. 

EXERCISES 

1.  If  6  times  any  number  whatever  be  added  to  13  times  that  number, 
the  result  is  how  many  times  the  given  number  ? 

2.  To  6  times  any  given  number  add  13  times  that  number,  and  to 
this  sum  add  -8  times  the  given  number ;  what  is  the  result  ? 

*  Since  18  +  (-  7)  =  11 ;  compare  §  16. 


28-30]  ADDITION  45 

3.  State  in  words  a  convenient  rule  for  adding  any  number  of  like 
terms.  Does  your  rule  apply  to  cases  in  which  some  of  these  terms  are 
negative  ? 

4.  Find  the  sum  of  6  n,  7  n,  —  3  n,  18  n,  and  —  11  n. 

5.  Find  the  sum  of  4  a^x^,  5  a^x%  -  2  a^x%  and  -  6  a^x^. 

Simplify  the  following  expressions,  i.e.,  unite  similar  terms : 

6.  3  mxy^  +  (  -  4  mxy^)  +  (  -  12  mxy^)  +  5  mxy^ 

7.  14  ahx^  +  32  abx^  +  (  -  19  abx^)  +  5  abx^. 

8.  3  mp^  +  7  mp^  +  13  a^x  -  inip'^  +  (- 6 a%)  -  2 a%. 

9.  4:(a  -  b)  +  '6 (a  -  b)  -  2{a  -  b)  +  (a  -  b). 

10.  4  (aa:)^  +  11  (axy  -  3  (axy  +  [-  6  («x)2]. 

11.  7  (:r  +  2/  +  2)  +  19  (x  +  y  +  z)  +  4  (a;  +  ^  +  2:)  -  8  (a:  +  r/  +  2). 

12.  -  15  (ax^  +  3)  +  27  ^ax^  +  3)  -  9  (ax^  +  3). 

Add  the  following  terms,  uniting  as  far  as  possible,  and  indicating  the 
addition  where  necessary : 

13.  Smp%  -  8  mp%  5  a%  -  4  mp%  -  Sa%  and  2  a^x. 

14.  23  a\  5  &2,  _  8  a^b%  -  13  &2,  24  02^2,  and  -  19  a^. 

15.  -  5 (a  -  &),  2  (ax) 2,  -  8  (ax)^,  12  (a  -  J),  and  -  4  (aa:)2. 

16.  16  X,  —  y,  4:x,  —  X,  4  z,  5  ?/,  x,  2  x,  and  —32. 

17.  7nxy  +  nxz/  equals  how  many  times  xy  ? 

18.  ax2  +  &x2  —  cx2  —  Zx2  is  how  many  times  x2  ? 

30.  Addition  of  polynomials.  The  explanation  given  in  §  29 
for  the  addition  of  monomials  is  easily  extended  so  as  to  apply 
to  the  addition  of  polynomials  also. 

E.g.,  7  62^3  _  3  ax2  +  6  abc  and  4  bhj^  +  5  aa-2  — 12  ahc  may  be  added  thus  • 

7  62?/3_3ax2+  6  abc 
4  b'^ij^  +  5  0x2  _  12  abc 


1162«/8  +  2ax2—  6a6c 

Similarly  in  general,  hence: 

To  add  two  or  more  polynoTnials,  uurite  theirv  under  one 
anotlier  so  that  similar  terms  shall  stand  in  the  same 
column,  and  then  add  each  column  separately  as  in  §  29. 


46  ELEMENTARY  ALGEBRA  [Ch.  IV 

EXERCISES 
Find  the  sum  of  each  of  the  following  groups  of  polynomials : 

1.  6a-5&  +  3c,     7a  +  106-6c,     8a-9&-10c,    and    19a+8&+2c. 

2.  2c-l  d+Qn,    8c?-3n-9c,    4(/+16n-4c,    and   3c-4n  +  (i. 

3.  2  c  -1  d  -  X  +  Qn,  8^-14n-3z,  18z  +  10n  +  8c?+3a, 
4n-18c-5x  +  6rf,     19c  +  4a;  +  8n-6fi?,    and    5c  +  2^-10c-4^. 

4.  2  x8  +  7  6a;2  -  4  fi^x  +  3  &3,  8  ft^^  -  15  hx^  _  5  63  _  10  a-^,  3  a;^  -  6  fix^, 
46a;2  -  6  ?>3  +  10x8,   and    -hx'^-\-  x^-^ b». 

Simplify  the  following  polynomials,  i.e.,  unite  their  similar  terms : 

5.  8  ma;  -  5  x2  +  3  m2  +  2  a:2  -  8  m2  +  13  m2  -  18  W2X  +  6  a;2  -  9  m^. 

6.  3a2-6a6-862  +  7a2_3a2  +  2a6-14&2_6a6  +  862. 

7.  4x2^-0:?/+  10x3-4i/2_8x8-4a:3-f  3?/2_  15xy  +  2'dxy. 

a   4  a2  -  6  a  +  4  -  3  a2  +  a  +  1.5  a2  -  2  +  5  a  -  3.4  a2  _  3.75  -  2  a. 
9.   ax^  -  4  a;2  +  &?y3  _  ^^2  +  14  x^  -  by^  +ay^  -3  y^. 
[Collect  all  the  x^  terms  and  all  the  y^  terms.] 

II.     SUBTRACTION 

31.  Subtraction  of  monomials.  Since  5  times  any  given  number, 
minus  2  times  that  number,  is  3  times  the  given  number,  i.e., 
(5—2)  times  the  given  number,  therefore  5  a— 2  a  =  (5  —2)  a =3  a, 
whatever  the  number  represented  by  a.  So  too  13  mx^y^  —  Smci^y^ 
=  (13  —  8)  mx^y^  =  5  mx^y^. 

Observe  that  the  reasoning  just  now  given  applies  to  any  two  similar 
monomials  whatever,  hence: 

To  subtract  one  of  two  sivxilar  monomials  from  tl%e  other, 
subtract  the  coefficient  of  the  subtrahend  from  that  of  tl%e 
minuend,  and  to  this  reinainder  annex  the  com^mon  literal 
faxitors. 

Here,  as  in  arithmetic,  it  is  usually  most  convenient  to  write  the 
subtrahend  under  the  minuend,  thus : 

126  a^s  13ma;2?/8  53  6ex8 

92fl22  8  rna;2ji/8  —  9  hcx^ 

34  a  22  5mx^y^  62  6car8* 

*  Since  63  —  (—  9)  =  53  +  9  =  02 ;  compare  §  17. 


30-31]  SUBTRACTION  47 

Note.  Since  algebraic  expressions  represent  numbers,  the  rule  just  now  given 
may  be  stated  thus : 

To  siibtract  one  of  tioo  similar  monomials  fro7n  the  other,  reverse  the  quality 
sig7i  of  the  subtrahend  and  proceed  as  i7i  addition  (ef .  §  17,  Ex.  5). 

In  order  to  avoid  confusion  when  reviewing  one's  work,  it  is  usually  best  not 
actually  to  change  the  sign  of  the  subtrahend,  but  only  to  conceive  it  to  be 
changed,  or  at  most  to  write  the  changed  sign  helovo  the  term,  thus: 

13  77ixhj^  53  6cx3 

8  mxhj^  —  9  6cx8 

+ 


5  mx2«/8  62  6ca;8 


EXERCISES 


In  the  following  exercises  subtract  the  number  written  below  from  tlie 
one  above  it : 


I" 


1. 

2. 

^8^ 

I.    1 

7a 
4a 

locx^ 
^cx- 

-18 
5 

16  6^2 
-3  6x2 

6  m'^p'^ 
—  5  m-^9* 

-18 
-5 

-  18  m^ 
b7n^ 

18                -9 
-5                    9 

-  18  rH^ 
-5r2a;8 

-  34.7  k^Y 

6.8  k'^xY 

9 

-9 

26  vY 

-  7  vh/ 

3. 

- 

5|  a2w?* 
-  2\  a"'m^ 

4.  Are  the  signs  written  in  the  above  exercises  signs  of  operation 
or  signs  of  quality  ? 

5.  Define  subtraction,  and  from  your  definition  show  how  to  verify 
the  correctness  of  the  above  exercises. 

6.  Show  that  "  changing  the  sign  of  the  subtrahend  and  proceeding 
as  in  addition  "  will  give  the  remainder  in  each  of  the  above  exercises. 

7.  From  5 (a  — 2  6^)  subtract —11  (a  — 2  6-^)  ;  also  subtract  15  m^(x—y) 
from  —  23  7n^(x  —  y)  ;  and  —  2  x{l  +  5  a2y/)  from  14  x(l  +  5  a2^). 

8.  From  the  sum  of  Q  ax^,  —  3  ax^,  and  11  ax^,  subtract  the  sum  of 
—  4  ax^,  9  ax^,  and  —  7  ax^. 

9.  Re-read  §§  16  and  17,  and  then  pi-ove  that,  in  any  subtraction,  the 
remainder  may  be  obtained  by  adding  the  subtrahend,  with  its  sign 
changed,  to  the  minuend. 


48  ELEMENTARY  ALGEBBA  [Ch.  IV 

32.  Subtraction  of  polynomials.  From  the  reasoning  already 
given,  it  is  evident  that  one  polynomial  may  be  subtracted  from 
another  by  writing  the  subtrahend  under  the  minuend,  similar 
terms  under  one  another,  and  subtracting  term  by  term,  thus : 

4  62y8  +  5  aa.2  _  12  ahc 
-       -  + 

352y3_8aa;2  +  i8a6c 

EXERCISES 

^1.  From  12  a  —  3  &  subtract  6  a  —  5  &. 

—2.  From  3a:— 2y+53  subtract  5 y  —  2  —  8 ar. 

^-Q.  From  4  a^xi/  -9x'^y  +  10  a^y'^  take  7  a32/2  _  3  a^xy^  -  12  x2?^. 

--4.  From  8  ^2  _  7  y^iS  _  13  aa;2  take  4  m^  -  8  aa;2. 

-5.  From  5  a;2  +  4  a%^  take  13  a^"^  -2x^+5  abx. 

6.  From  a;^  +  1  take  1  -  2  x  +  a;^  +  3  a;2  -  4  a;3. 

7.  From  2  a  —  3 x+  z  take  the  sum  of9a;  +  z  —  4a  and  10 z—  5 x+  a. 

8.  From  7.42  a:2  -  3}  xy  +  10  y'^  take  2.5  xy-7f-\-  3.02  a,-2. 

9.  From  34  a2x8  -  10  mV  take  15  if  +  10  a2a:3  +  m*y\ 

10.  Subtract  -  7  c^r^  +  3  a2  _  ^s  from  5  a2  +  2  r2  +  .s*  -  3  ch^ 

11.  Subtract  1  -  3  a:  +  10  a:2  from  2  a;2  +  5 ;  from  4 ;  from  0. 

12.  Subtract  -8a  +  3  &-13x2  from  5  6;  from  -6x2  +  2 a;  from  -7. 

13.  Subtract  3  &2  __  10  aa:  +  5  a:2  from  the  sum  of  5  aa:  —  2  a:2  and 
10  62  _  13  ax. 

14.  Subtract  the  sum  of6a  —  46  +  3c  and  6h  —  2a  —  c  from  8 a  —  3 6. 

15.  Subtract  3  a:  -  10  ay^  —  2a^  minus  x  —  6ay+a^  from  4  a;2  —  a;  plus 
5  a3  -  3  ay\ 

16.  From  fa:8-|a;2+|a;-3  subtract  |  a:^  -  2^  -  ^  a:  -  f  x^. 

17.  Subtract  1  +  3(x  -  y)  -  5(a2  +  b)  from  a^  +  2(a^+b)-  8(x  -  y). 

18.  Subtract  the  sum  of  5  a  -  3  &2  +  2  x  and  -  4  x  +  2  ^2  from 
3  x2  +  4  a  -  12  52  minus  3  a  -  7  62  +  2  x2. 

19.  Subtract    4.5  m  -  1.3  y'  +  10  a^c^    from    1.4  y  -  8  a^c^  plus    6.3  y 

-  181.  ^2^4. 

20.  From  the  sum  of  x2  -  1 ,  3  x  +  2,  and  -  8  x2  -  5  x,  subtract  4  x  -  3  x2 
plus  4  —  2  X  minus  6  x  +  3  x2  —  8. 


> 


32-34]      ALGEBliAlG  EXFUESlSlONS  —  PARENTHESES  49 

III.     PARENTHESES 

33.  Removal  of  parentheses.  That  one  expression  is  to  be  sub- 
tracted from  another  may  be  indicated  by  inclosing  the  subtra- 
hend in  a  parenthesis  and  writing  the  minus  sign  before  it. 

E.g.,  6  X  —  (2  a  —  ?/)  means  that  2  a;  —  r/  is  to  be  subtracted  from  6  x. 

Moreover,  since  a  subtraction  is  performed  by  changing  the  sign 
of  each  term  of  the  subtrahend  and  then  adding  it  to  the  minuend 
(§§  32  and  31),  therefore  a  parenthesis  preceded  by  the  minus 
sign  may  he  removed  by  simply  changing  the  sign  of  each 
term  inclosed  by  it* 

E.g.,  a—{—  be  +  mp)  =  a  +  bc  —  mp;3kx2~i2by  —  7a2)=3kz^  —  2by  +  7a^', 
a;2  +  2  6x  —  (&3 ~  bx  +  S x^)  =  x^ -\- 2  bx  —  b''^ -\-  bx  —  3 x^  =—  2x'^-^Sbx  —  b^;  and 
—  (-  4  A;2  +  5  ax  -  8  6?/3)  =  4  ^^  —  5  ax  +  8  by^. 

Note.  If  a  parenthesis  is  preceded  by  the  plus  sign,  it  may  be  removed  without 
changing  the  signs  of  the  terms  inclosed  by  it,  because  the  expression  within  such 
a  parenthesis  is  to  be  added  to  whatever  precedes  it. 

34.  Parenthesis  within  parenthesis.  It  often  happens  that  a 
sign  of  aggregation  may  inclose  one  or  more  other  signs  of  aggre- 
gation, thus : 

3a''x-l2  mb  +  [a^x  _  (-  4  s^^  -f  5  mb)  -f  sH']  J . 

In  such  cases  it  may  be  best  for  the  beginner,  after  removing 
all  those  signs  of  aggregation  which  are  preceded  by  the  plus 
sign  (§  33,  note),  to  remove  the  innermost  of  those  signs  of  aggre- 
gation which  are  preceded  by  the  minus  sign,  then  the  next  inner- 
most, and  so  on  until  all  are  removed. 

E.c/.,  omitting  the  square  bracket  in  the  above  expression,  since  it  is  preceded 
by  the  plus  sign,  that  expression  becomes 

3 a2x -{2mb  +  a^x -  (- 4 s2i  +  5 mb)  +  sH}; 
now  removing  the  parenthesis,  this  expression  becomes 

3  a2x  -  (2  m6  +  a2x  -h  4  s^t  —  5  m&  4-  s^] ; 
and,  removing  the  brace,  we  obtain 

3  a2x  —  2  mb  —  a2x  —  4  ^2^  +  5  mb  —  s% 
i.e.,  2a2x  +  3m6  — 5s2<. 

*  Compare  also  §  39,  Ex.  19. 


50  ELEMENTARY  ALGEBRA  [Ch.  IV 

Note.  The  work  of  removing  parentheses  in  such  expressions  as  that  just 
given  may  be  somewliat  shortened  by  removing  the  outermost  negative  paren- 
thesis first,  then  the  next  outermost,  and  so  on,  instead  of  beginning  with  the 
innermost.  The  expression  witliin  an  inner  parenthesis  is,  of  course,  to  be 
regarded  as  a  single  term  of  the  next  outer  parenthesis.  Parentheses  preceded 
by  the  sign  +  should  be  dropped  whenever  they  occur.  The  student  may  simplify 
the  above  expression  by  this  method  and  then  compare  his  work  with  that  above. 

The  essential  thing  in  both  plans  is  that  on  removing  a  negative  parenthesis 
the  sign  of  every  term  inclosed  by  it  must  be  reversed. 


EXERCISES 

Simplify  the  following  expressions : 

1.  7a;-3ac  +  (a:-2ac). 

2.  1 X —  Zac —{x —  2ac). 

3.  4  a  -  2  &  -  (c  +  3  a)  -  (2  c  +  3  &  -  2  a). 

4.  5  a:2  +  (7  aa:  -  10  ^/)  +  3  ?/  -  (4  ao;  -  5  y  +  3  a;2). 

5.  ^xy+^if-^-x^  +  y'^  +  xy). 

6.  mx2-[8?/+(6a-wa:)-2a]. 

7.  _  (a  +  5  -  c)  +  4  a  -  (c  +  3  h). 

8.  ^x-2y+f-g-{2x-{^y  +  ^z-2V)^-2f-2g]. 

9.  a-y-{a-{-y-^^r^)}. 

10.  15  -  (6  -  a:)-  [13  -  {x-{y  +  2)  +  2  ?/}  +  2  x]. 

11.  x-{Zx-l-{-^x  +  2y)+by^-^y]. 

12.  -{-[-(x-y)]}. 

13.  8  a  -  2  &  -  {(3  6-  -  <f)  -  [4  c  -  rf  -  (-  8  a  +  2  &)]  -  2  c?}. 


14.  4  -  [5  1/  -  {3  -  (2  a;  -  2)  -  4  a;}]  -  {x  +  5  1/  -  X  +  3}. 

15.  5  a^xV  -  {2  a^xY  -  ["^  +  (3  x'^y'^  -a^-^  a'^xY)  +  4  a-]  -  3  xY}- 


16.  a'»-n''-(3a»-2n«)  +  (-  5  a'»  -  2n«)-{-  [-(-  «"-n«)]}. 

17.  -  d  ax  -  (o  xy  -  ^  z)  +  2  z  -  1(4:  xy  +  6  z  +  ax)  +  3  xy']. 

18.  2  a -la -{h -(3b-  2a-b)  -  3  a}  +  4  ^»]  -  (6  -  a). 

35.  Inserting  parentheses.  From  §§33  and  34  it  follows  that 
the  value  of  a  polynomial  is  not  altered  by  inclosing  any  number 
of  its  terms  in  a  parenthesis,  provided  only  that  if  this  paren- 
thesis is  preceded  by  the  minus  sign,  the  sign  of  each  inclosed 
term  be  reversed. 


34-35]     ALGEBRAIC  EXPRESSIONS  —  PARENTHESES  51 

EXERCISES 

1.  Indicate  by  means  of  a  parenthesis  that  a  +  b  -\-  c  is  to  be  sub- 
tracted from  a  —b  -\-  c;  then  remove  the  parenthesis  and  simplify  the 
expression. 

2.  Inclose  the  last  two  terms  of  x^-\-  y^  —  z^  in  a  parenthesis  preceded 
by  the  plus  sign ;  by  the  minus  sign.* 

3.  Inclose  the  last  three  terms ofax  —  4?/4-3a  —  8a:ina parenthesis 
preceded  by  the  plus  sign;  by  the  minus  sign. 

4.  In  the  expression  3w  —  4a  +  10a;2— 5^  +  3  ab^  —  8  aa:,  inclose  the 
4th  and  5th  terms  in  a  parenthesis  preceded  by  the  minus  sign ;  then 
inclose  this  parenthesis,  together  with  the  two  preceding  terms,  in  a 
bracket  preceded  by  the  minus  sign. 

SO.  Make  the  clianges  asked  for  in  Ex.  4,  in  the  expressions 
3'^  +  4  a  -  10  a;2  -  5  2/  +  3  a&2  _  8  aa:,  3  m  -  4  a  -  10  a:2  +  5  jr  -  3  0^2  +  8  rta:, 
and  -  5  x2  +  3  2/2  -  4  a  -  14  Jc  +  8  m\ 

6.  Inclose  the  first  three  terms  of  each  of  the  expressions  in  Ex.  5  in 
a  parenthesis  preceded  by  the  plus  sign ;  preceded  by  the  minus  sign. 

7.  When  terms  are  inclosed  in  a  parenthesis  preceded  by  the  plus 
sign,  are  any  changes  in  the  signs  of  these  terms  made?  Why  ?  Explain 
why  the  signs  are  changed  when  the  parenthesis  is  preceded  by  the  minus 
sign. 

8.  Just  as  5  x  +  3  x  =  (5  +  3)a:,  so  ax  -{■  bx  =  {a  ^  y)x,  and  similarly, 
mx  —  nx  ■\-  px  =  {jn  —  n  -{■  p)x  =  — (— w  +  n—  p)x. 

Similarly  combine  the  terms  oibx  —  mx  —  nx. 

9.  Combine  all  the  a:-terms,  and  also  all  the  y-terms,  in  the  following 
expressions :  ax  —  bi/  —  dy  —  ex—  ex  -\-fy,  mx  —  ex  ■\- py  —  ay  +  gx,  and 
d  ex  +  4:  dy  —  2  ax  —  5  7nx  —  7  by  -^  ax. 

10.  Arrange  the  letters  within  the  parentheses  in  the  expressions  of 
Ex.  9  in  their  alphabetical  order,  and  give  to  each  parenthesis  the  sign 
of  the  first  letter  it  contains. 

11.  Group  together  the  like  powers  of  y  in  the  following  expressions : 
ay^  —  2  by  —  3  ey^  —  my^  —  ny  +  dy%  y^  —  ay^  —3ry^-\-  ny^  —  ly^,  and 
-Sy^-ci/  +  ay-  dy  ■^by^-2  ay^  +  ny"^  -  y. 


*  In  such  exercises  it  is,  of  course,  understood  that  the  value  of  the  expression 
is  to  be  left  unchanged. 


CHAPTER   V 

MULTIPLICATION  AND  DIVISION  OF  ALGEBRAIC 
EXPRESSIONS 

I.    MULTIPLICATION 

36.  Some  fundamental  laws.  Before  going  farther  it  is  perhaps 
well  to  point  out  that  thus  far  in  this  book,  as  well  as  in  the 
arithmetic  previously  studied,  it  has  been  silently  assumed  that, 
whatever  the  numbers  represented  by  a,  h,  and  c, 

a  +  h  +  c  =  a  +  c  -\-h  =  h  -\-  c-\-ay  etc. ; 

i.e.,  it  has  been  assumed  that  the  sum  of  several  numbers  is  not 
changed  by  changing  the  order  in  which  these  numbers  are  added. 
This  is  known  as  the  commutative  law  of  addition. 

This  assumption  was  based  upon  the  fact  that  with  any  par- 
ticular set  of  numbers,  such  as  2,  5,  and  8,  the  correctness  of  these 
statements  (equations)  is  easily  verified. 

E.g.,  2  +  5  +  8  =  2  +  8  +  5.  [Each  member  being  15 

It  has  also  been  assumed  that  the  sum  of  several  numbers  is  not 
changed  by  grouping  together  any  two  or  more  of  the  summands. 
and  replacing  them  by  their  sum.  This  is  known  as  the  associa- 
tive law  of  addition. 

E.g.,  a  +  6  +  c  =  a  +  (6+c). 

The  commutative  and  associative  laws  of  multiplication  are  ex- 
pressed by  such  equations  as 

and  a  •  h  '  c  —  a  '  {h  •  c), 

respectively;  their  correctness  has  also  thus  far  been  assumed. 

While  attention  is  now  expressly  called  to  the  fact  that  mere 
verifications,  however  numerous,  cannot  prove  the  generality  of  a 
law,  the  proofs  of  the  above  laws  are  deferred  till  Chapter  VI ; 
mitil  then  their  correctness  will  continue  to  be  assumed. 

62 


36-38]  MULTIPLICATION  68 

37.  Law  of  exponents  in  multiplication.  The  words  "power," 
"  base,"  and  "  exponent,"  as  used  in  connection  with  arithmetical 
numbers,  were  defined  and  illustrated  in  §  7  (iv).  The  definitions 
there  given  apply  also  when  algebraic  numbers  are  under  con- 
sideration, though  it  is  to  be  carefully  noted  that,  while  the  base 
and  the  power  may  be  negative  or  fractional,  the  exponent  (under 
the  present  definition)  is  necessarily  a  positive  integer. 

It  follows  directly  from  these  definitions  that,  if  a  represents 
any  number  whatever,  then 

a^ .  a^  =  (a  •  a  •  a)  •  («  •  a)  =  a  •  a  •  a  •  a  •  a     [Associative  law 

=  a\ 

i.e., 

Similarly  in  general,  if  m,  n,  and  p  are  any  positive  integers 
whatever,  then 

a"* .  a"  =  (a  •  «.  •  a .  •••  to  m  factors)  -{a -a-  a-  •••  to  n  factors) 

^a-  a-a-  •••to  (m  +  w)  factors  [Associative  law 

=  a'"+". 
So,  too,  a"^  -a""  -a^  =  a"*+"+^. 

The  law  of  exponents,  expressed  by  these  equations,  may  be 
formulated  into  words  thus  :  the  product  of  two  or  more  powers 
of  any  numher  is  that  power  of  the  given  ninnber  ivhose 
exponent  is  the  sum  of  the  exponents  of  the  factors. 

38.  Product  of  two  or  more  monomials.  The  product  of  two  or 
more  monomials  may  be  obtained  as  a  simple  extension  of  §  37. 

E.g.,  if  a,  b,  and  x  represent  any  numbers  whatever,  then 

(2  ax3)  .  (3  b^x)  =  2  •  a  •  x3  •  3  •  62 .  a;  [Associative  law 

=  2  •  3  •  a  •  62 .  x3 .  X  [Commutative  law 

=  6  ab^xA.  [Associative  law 

Similarly,    (3  a^^)  •  (-2  a6x2)  •  (5  a62x4)  =3  •  (-2)  •  5  •  a2  •  a  •  a  •  6  •  62  •  x3  •  a;2 .  x4 

=  -30a463x9. 

And,  manifestly,  the  product  of  any  number  of  monomials  may  be  obtained  in 
the  same  wav. 


54  ELEMENTARY  ALGEBRA  [Ch.  V 

This  method  of  obtaining  the  product  of  several  monomials  may 
be  formulated  into  the  following  rule :  to  the  product  of  the 
numerical  coefficients  of  the  several  monomials,  annex  each 
of  the  letters  which  they  contain,  and  give  to  each  letter  an 
exponent  equal  to  the  sum  of  the  exponents  of  that  letter 
in  the  several  monoinials. 

EXERCISES 

1.  Define  and  illustrate  the  meaning  of  exponent,  base,  and  power. 

2.  May  the  base  be  a  negative  number?  a  fraction?  May  the 
exponent  be  either  negative  or  fractional  ? 

3.  If  the  base  is  a  fraction,  what  is  the  power?  If  the  base  is  nega- 
tive and  the  exponent  is  3,  is  the  power  positive  or  negative  ?     Why  ? 

4.  If  the  base  is  negative,  what  is  the  sign  of  the  power  when  the 
exponent  is  4?  when  it  is  5?  when  it  is  6?  when  the  exponent  is  even? 
when  it  is  odd? 

5.  What  is  the  meaning  of  a:^?  of  x^?  How  many  times  is  x  used 
as  a  factor  in  a:^  •  a;^?  How  then  may  this  product  be  represented?  State 
the  law  of  exponents  for  multiplication. 

6.  If  X  stands  for  a  negative  number,  is  x^  positive  or  negative? 
Why?  How  does  3*  compare  with  (-3)^?  2^  with  (-2)6?  2^  with 
(—2)5?     State  the  general  law  of  which  these  are  particular  cases. 

7.  What  is  the  meaning  of  a^y^'^  of  a^^^?  How  many  times  is  a  used 
as  a  factor  in  the  product  a^y^  •  a^y'^'^  How  many  times  is  y  so  used?  In 
what  simpler  form  may  this  product  be  written  ?     Why  ? 


Multiply 
by 

8. 

4  n^x^ 
3ax2 

13. 

7p2,p3y5 

-  9  pw'^x^ 

9. 

2m8<2 

10. 

-  8  an  V 
5ay^ 

11. 

-4a2j3^2 

-  6  a^hx 

15. 

xy^z^ 
-  xhjh 

12. 

5  b^x^ 
-  7  a^by^ 

Multiply 

by 

14. 

m^x^ 
-  3  p^xy^ 

16. 

f  ahn'^x^ 
-  f  bmY 

17.  Write  a  carefully  worded  rule  for  finding  the  product  of  two 
monomials  —  it  should,  of  course,  make  special  mention  of  the  coefficient, 
the  letters,  the  exponents,  and  the  sign  of  the  product. 

18.  Find  the  product  of  4  ax%  -  2  a^xy\  and  5  aby^. 

19.  What  is  the  product  of  3  m^pw\  —2  ap^w^,  —  6  mp"^,  and  —  aw^l 


38-39]  MULTIPLICATION  66 

20.  Wliat  is  the  product  of  2^  ab%  1.2  b^x%  and  -  f  a%  ? 

21.  What  is  the  meaning  of  a:«?  May  x  represent  any  number  what- 
ever here?  may  w?  How  may  the  product  of  a;«  and  x^  be  represented? 
of  a^,  a"*,  and  a''?  What  are  the  restrictions  upon  m  and  r  in  this  last 
question  ? 

22.  What  is  the  meaning  of  ?/"-2?  What  are  the  limitations  upon 
n  here?  What  is  the  product  of  4  a^  and  -3a«-2?  of  2  a'"^"  and 
—  a'^-^x*?  Does  the  answer  given  to  Ex.  17  apply  to  such  multiplica- 
tions as  these? 

23.  What  is  meant  by  (a^)*?  by  (a;8)2?  by  (-3a2?/)2?  W>ite  each 
of  these  expressions  in  its  simplest  form. 

24.  Without  actually  performing  the  following  indicated  operations, 
tell  by  inspection  what  the  sign  of  the  result  is  in  each  case,  and  why : 
(_3)4.  (-2)9;  (-11)40;  5^  724;  (_5)«when  n  is  an  even  positive 
integer,  and  when  n  is  an  odd  positive  integer;  (— 3)2«and  (— 3)2«  +  i, 
when  n  is  any  positive  integer. 

25.  As  in  Ex.  24,  determine  the  sign  of  the  result  in  each  of  the  fol- 
lowing indicated  operations  if  a  =  2  and  6  =—4:  (a  — 6)^;  (a  — 6)*; 
(a+by-,  {ab'^Y]   (a-4:by;   {o.^'^y-,  and  {a%y^\ 

26.  Tell  what  is  meant  by  the  commutative  and  associative  laws  of 
addition  and  multiplication.     Illustrate  your  answer  in  each  case. 

39.  Product  of  a  polynomial  by  a  monomial.  Since  the  product 
of  two  numbers  is  obtained  from  the  multiplicand  in  the  same 
way  as  the  multiplier  is  obtained  from  the  positive  unit  [§  3  (iii)], 
therefore  5  •  (2  +  6)  =  5  •  2  -f  5  •  6,  because  the  multiplier  2  -f  6  is 
obtained  by  first  taking  the  unit  2  times,  then  6  times,  and  adding 
the  two  results. 

Similarly,  whatever  the  numbers  or  expressions  represented 
by  a,  b,  c,  d,   ", 

a{b-\-  c  +  d  -\-  •••)  =  ab  -\- ac  -\-  ad  +  ••• ; 

and,  applying  the  commutative  law  to  each  member  of  this  equa- 
tion, it  becomes 

(b-\-c  +  d-\ )•  a  =  ba  +  ca-\-da  +  ••♦. 

These  last  two  equations  state  what  is  known  as  the  distributive 
law  *  of  multiplication  as  to  addition ;  it  may  be  put  into  words 

*  The  multiplication  of  a  sum  is  "  distributed  "  over  the  parts  of  that  sum. 


56  ELEMENTARY  ALGEBRA  [Ch.  V 

thus:  the  product  of  a  polynomial  hy  a  jnonomial  is 
obtained  hy  multiplying  each  term,  of  the  polynomial  hy 
the  monomial  and  adding  the  partial  products. 

E.g.,  5a;(3a2-2  6  +  c2)  =  (3a2-26  +  c2)  .  5x  =  16 a2a;-10 &x  +  5 c2a;.      The 
actual  work  may  be  conveniently  arranged  thus: 

3  a2  -  2  6  +  c2 
5x 


15  a2x  —  10  6x  +  5  c^x. 


each  term  of  the  multiplicand  being  multiplied  by  the  multiplier,  and  the  partial 
products  added. 

EXERCISES 

1.  How  is  a  +  1  —  c  obtained  from  +  1  ?  How  then  is  the  product 
3  •  (a  +  6  -  e)  to  be  obtained  from  3  ? 

2.  Is  3  •  (a  +  &  -  c)  equal  to  (a  +  &  -  c)  •  3  ?    Why? 

3.  What  is  the  product  of  365  by  2?  of  (300  +  60  +  5).  2?  Show 
that  this  illustrates  the  distributive  law. 

4.  Since    a(6  ■{■  c  ■\-  d  -^  •••)  =  (6  +  c  +  c?  H — )  '  a  =  ah  +  ac  +  ad-\ , 

whatever  the  numbers  represented  by  a,  b,  c,  d,  •••,  what  is  the  product  of 
2  ax  and  3  a:2-4  a2a;8  +  5  aa;*? 

i> 

5.  Multiply  3  a'^b^—7  ax  by  2 abx.     Also  5 mx^ -7ay^-4:  aHi  by  - 2  am^. 

Write  a  rule  for  multiplying  a  polynomial  by  a  monomial. 

6.  When  an  indicated  multiplication  has  been  performed,  and  the 
result  is  expressed  by  an  equation,  is  that  equation  an  identity  or  merely 
a  conditional  equation  ?  E.g.,  is  (3  a%^  —  7  ax)  •  2  abx  =  6  a%^x  —  14  a^x^ 
a  conditional  equation  or  an  identity  ? 

7.  The  fact  that  the  equation  in  Ex.  6  is  an  identity  may  be  used  as  a 
partial  check  upon  the  correctness  of  the  multiplication.  Are  the  two 
members  equal  when  a  =  &  =  a;  =  l?  If  they  were  not  equal  when  these 
special  values  are  assigned  to  the  letters,  could  the  multiplication  be 
correct  ?  Does  the  equality  of  the  two  members  for  this  set  of  values 
prove  that  the  multiplication  is  correct,  or  does  it  merely  increase  the 
probability  of  its  correctness?  Is  it  then  a  "  complete  "  or  only  a  "  partial " 
check  ? 

8.  9.  10. 

Multiply      8a2-4aa:  +  3m2  -3  xh -5  x^ +  4:xz^  2a-3J  +  c 

by  —  4  am^  —2xz^  '  —  abc 


o'J-40]  MULTIPLICATION  67 

11.  Check  Exs.  8,  9,  and  10  by  the  method  of  Ex.  7.  Could  other 
special  values  for  the  letters  than  those  there  given  be  employed  for  such 
a  check?     Why? 

Multiply  (and  check  the  work) : 

12.  5  m2  -  2  P  by  3  mF.  13.    -  8.5  h^x'^y  +  5f  hy^  by  ^V  ^y- 

14.  25  «»  -  17  a^  _  a6  by   -  3  a\ 

15.  a;^6  _  2  x^y^  —  15  a:*?/^  +  4  x^y  by   —  x^-'^y'^-^. 

Perform  the  following  multiplications  and  check  the  work: 

16.  -  2  a:2  .  {x^  -bxhj-  16  x'^y'^  +  24  xy^  -y^-xy-^). 

17.  (a362c3  _  3  ah^c^  -  4  a^fes^  +  aj^)  •  2  a&c^. 

18.  -1.  (3r«a;-4m2-2a;2). 

19.  Since  -1.(3  mx  -  4  m^  -  2  a:^)  =  -  (3  mx  -  4  m^  -  2  a:2),  derive 
from  P]x.  18  a  new  proof  that  a  parenthesis  preceded  by  the  minus  sign 
may  be  removed  if  the  sign  of  each  term  inclosed  by  it  be  reversed 
(cf.  §  33). 

40.  Product  of  two  polynomials.  Since  m  +  n  is  obtained  from 
the  positive  nnit  by  adding  n  times  this  unit  to  m  times  the  unit, 
therefore,  by  the  definition  of  multiplication, 

(a  4-  6  +  c)  •  (m  +  n)  =  (a  +  6  +  c)m  +  (a  +  6  +  c)n 

=  am  +  6m  +  cm -\-an-\-hn  +  en.       [§  39. 

Similarly  for  any  polynomials  whatever;  i.e.,  the  product  of 
two  polynomials  is  obtained  hy  multiplying  each  term  of 
the  multiplicand  hy  each  term  of  the  multiplier,  and  add- 
ing the  partial  products. 

If  any  two  or  more  terms  of  a  product  are  similar,  they  should, 
of  course,  be  united. 

The  actual  work  of  such  a  multiplication,  and  its  check,  may  be  conveniently 

arranged  thus:  Check 

a2  +  2  a&  -  62  =  +  2,  when  a  =  6  =  1 

a+6 =  +2,  when  a  =  6  =  1 

(a2  +  2a&  — 62)  .a=  a3  +  2a26-a62 

(a2  +  2  a6  -  62)  .  6=  a26  +  2a62— 6^ 

a8  +  3a26  +  a62-68  = +4,  when  a  =  6  =  1 

Note.  The  product  of  three  or  more  polynomials  may  be  obtained  by 
multiplying  the  product  of  the  first  two  by  the  third,  this  product  by  the  fourth, 
and  80  on. 


58  ELEMENT ABY  ALGEBRA  [CJh.  V 

EXERCISES 

Multiply  (and  check  the  work)  : 

1.  4  az  +  5  a2  -  2  a;2  by  3  a  -  4  a:. 

2.  2x^-7  xy  +  Sa^x  hj  -  5  x  +  3  y. 

3.  4  m2  -  3  mp  by  3  p"^  -  2  m  +  m^. 
V4.  5s-3^  by  2s  -Zr  +  t. 

5.  ax"^  —  hij'^  by  hx  +  ay. 

6.  a^  —  2  «:c  +  a;^  by  a  —  x. 

7.  2  a^  -  6  a6  +  3  &-2  by  a  +  6  +  aft. 

8.  X  -  5  x^  +^10  by  2  -  7  X  +  a;"^- 

9.  ai*  -  2  a^r  +  5  by  a  -  X  -  3. 

10.  w^  +  2  m.n  +  n^  by  m  +  n  —  mn. 

11.  a  +  6  —  c  +  c?  by  a  —  6 +  c  —  rf. 

12.  3  a  -  5  ^2  _^  a/x  by   -  f  +  2  a  -  3  x^. 

13.  a^  +  b'^  +  c"^  -  2  ah  -  2  ac  +  2hc  hy  a  -  h  -  c. 


f 


14. 

xn  +  j,H   by   X  -  y. 

15. 

a:n  +  yn   by  :c2  _  ^2. 

16. 

X"  +  r  by  X"  -  2/«. 

17. 

X"  +  ?/»  by  x'-  -  J/*-. 

18. 

3  a8  -  4  a26  +  2  ab^ 

19. 

1.8x2-2x^-2.3  5 

20. 

2.5a2a;2_i,4a:ry  + 

ft3  by  5  a2  -  3  aft  +  ft2. 
by  lix-3|3/. 
1 3/2  by  -  3  ax  -42/  -  1.2  a. 

41.  Integral  expressions,  degree  and  arrangement  of  expressions, 
etc.  In  multiplications  with  polynomials,  and  elsewhere,  it  is 
often  advantageous  to  arrange  the  terms  of  a  polynom.ial  in  a 
particular  order;   such  arrangements  will  now  be  explained. 

A  term  is  said  to  be  integral  if  it  contains  no  letters  in  its 
denominator ;  *  it  is  integral  imth  regard  to  a  particular  one  of 
its  letters  if  that  letter  does  not  appear  in  its  denominator.  A 
polynomial  is  integral,  or  integral  with  regard  to  a  particular 
letter,  if  each  of  its  terms  is  so. 

E.g.,  3ax'^+-—^ — oct^y  jg  integral  with  regard  to  b,  w,  x,  and  y,  it  is 

fractional  with  regard  to  a;   its  first  and  last  terms  are  altogether  integral, 
while  its  second  term  is  integral  only  with  regard  to  6,  m,  and  y. 

*  It  may  contain  numerical  denominators  and  still  be  called  integral. 


40-42]  MULTIPLICATION  59 

By  the  degree  of  an  integral  term  is  meant  the  number  of 
literal  factors  which  that  term  contains,  i.e.,  it  is  the  sum  of  the 
exponents  of  all  the  letters  of  that  term. 

E.g.,  5  ax  is  of  the  2d  degree,  and  32  a^cy^  is  of  the  8th  degree. 

An  integral  polynomial  is  said  to  be  of  the  same  degree  as  its 
highest  term ;  if  all  of  its  terms  are  of  the  same  degree,  it  is  said 
to  be  homogeneous. 

E.g.,  6  aby-  —  2  bmx  +  5  ciH^y  is  of  degree  6,  and  2  ax^  —  6  xyz  +  5  abx  —  y^  is 
homogeneous,  and  of  degree  3. 

One  is  often  concerned  with  the  degree  of  a  polynomial  (or  of  a 
term)  with  regard  to  some  rather  than  all  of  its  letters ;  in  such  a 
case  only  those  letters  are  considered  in  determining  the  degree. 

E.g.,  5  a^x^y  —  3  ab^xy"^  +  2  x^  is  homogeneous,  and  of  degree  3,  with  regard  to 
the  letters  x  and  y;  it  is  of  degree  2  in  y  alone,  and  of  degree  3  in  x  alone,  and 
non-homogeneous;  its  degree  in  all  the  letters  is  7. 

A  polynomial  is  said  to  be  arranged  according  to  ascending 
powers  of  some  one  of  its  letters  if  the  exponents  of  that  letter,  in 
going  from  term  to  term  toward  the  right,  increase,  and  that  letter 
is  then  called  the  letter  of  arrangement;  it  is  arranged  according 
to  descending  powers  of  the  letter  of  arrangement  if  taken  in  the 
reverse  order. 

E.g.,  2  x3  —  5  ax^y  —  7  b^y^  +  3  m^y^  is  arranged  according  to  descending 
powers  of  x,  and  ascending  powers  of  y. 

42.   Multiplication  in  which  the  polynomials  are  arranged.      If 

each  of  two  polynomials  be  arranged  according  to  powers  of  some 
letter  which  is  contained  in  each,  then  their  product  will  arrange 
itself  according  to  powers  of  that  letter,  and  the  actual  multipli- 
cation will  take  on  an  orderly  appearance. 

E.g.,  to  get  the  product  of  7x  —  2x^  +  5  +  x^hYSx  +  4:X^  —  2,  arrange  the 
work  thus:  Check 

x3-2a;2+  Tx  +   5  =11,  whenx=l 

4.r2  +  3  3;  —   2 =   5,  when  x  =  l 

4  x5  —  8  a;4  +  28  xS  +  20  3-2 

3x*-   0  3;3  +  21x2+15x 

—2x3+    4a-2— 14  a;  — 10 

4x5  — 5x*  + 20x3 +45x2 -1-x— 10  =55,  when  x  =  1 


60  ELEMENTARY  ALGEBRA  [Ch.  V 


EXERCISES 

1.  Is  the  monomial  fa^z*  integral  or  fractional?    With  regard  to 

what  letters  is     "  ^  -^   integral?     With   regard   to   what  letters   is  it 
fractional  ? 

2.  What  is  meant  by  the  degree  of  an  integral  algebraic  expression? 
When  is  such  an  expression  said  to  be  homogeneous? 

3.  Arrange  the  expression  4 ax^  —  7 x^  A-  b x^  —  2hx  —  ^ a^  according  to 
descending  powers  of  x.  Also  according  to  ascending  powers  of  x.  Of 
what  degree  is  its  present  first  term? 

4.  Arrange  the  expression  3  x'^y^  +  xy^  —  8  x^y^  —  6  x^y"^  +  x^y  accord- 
ing to  descending  powers  of  x.  How  is  it  then  arranged  with  reference 
to  y  ?    Of  what  degree  is  this  expression  ?    Is  it  homogeneous  ? 

In  the  following  exercises  arrange  both  multiplier  and  multiplicand 
according  to  some  letter  contained  in  each,  then  multiply  and  observe 
that  the  product  has  a  corresponding  arrangement. 

Multiply : 

5.  6  a;2  -  2  +  5  a:  +  3  a:8  by  a;2  +  5  -  a:. 

6.  2a  +  a8  -  a^  -  1  by  4  -  a2  +  a. 

7.  3  a^x  -  4  ax"^  ■{■  x^  -  a^  by  a'^-ax-\-  x\ 

8.  3  xy^-  2/8-3  x'^y  +  x^  hj  -2xy-\-x^  +  y\ 

9.  a:2^2  _  y.yz  +  2/4  _  ^.s^  ^  a:*  by  x'^-\-xy-  y\ 

10.   4A2r-/ir2-A8+2r«  by  ^-2r. 

IJ..  In  the  product  of  two  homogeneous  polynomials,  one  of  degree  5 
and  the  other  of  degree  2,  what  is  the  degree  of  each  term?  Why? 
Is  then  this  product  homogeneous?  Show  that  this  consideration  may 
be  used  as  a  partial  check  upon  the  correctness  of  such  a  product. 
Compare  also  Exs.  7-10. 

12.  Find  the  product  of  ax"^  +  h'^x  +  a%,  a-\-h+  x,  and  a  —  x.  Should 
this  product  be  homogeneous  ?     Why  ? 

13.  Find  the  product  of  2  m^  —  5  mn  +  3  n^,  3  m  —  2  n,  and  1  —  m  —  n. 
Should  this  product  be  homogeneous? 

Expand,*  and  check,  the  following  indicated  multiplications : 

14.  (Za  +  2h)(2ax-a'^-x'^)(hx-2a). 

15.  {x^  _  3  a:8^  +  2/4  -  3  xy^)  {x'^  -2xy+  y^). 

*  An  indicated  product  is  said  to  have  been  expanded  when  the  multiplication 
has  been  performed. 


42]  MULTIPLICATION  61 

16.  (3r2_5r  +  25)(s-2f  +  r)(3-s- 0- 

17.  [3  X  +  2  2/  -  3  (y  +  2  a;)  -  2]  [2  -  5  (a:  -  2  +  3y)]  (a;  +  y  -  1). 

18.  (x  +  yy,  i.e.,  (x  +  7/)(x  +  y)  (x -\- y). 

19.  (^x  -  yy(x -h  yy. 

20.  (a-2  6)8(2a-&)(2a  +  6). 

21.  (x2  +  xy  +  ?/2)(a:-2/). 

22.  (a8  +  a%  +  a6'^  +  63)  (a-b). 

23.  (z2  +  a:?/  +  ?/2) (x-2  _  a;e,  +  ^2)  (^  _  y)  (a;  4. 2,). 

24.  If  the  multiplier  and  multiplicand  are  each  arranged  according  to 
the  descending  powers  of  some  particular  letter,  how  will  the  product 
arrange  itself  ?  From  what  two  terms  is  the  highest  term  in  the  product 
obtained?     The  lowest  term  in  the  product? 

25.  The  results  in  Exs.  21-23  show  that  some  of  the  terms  of  a 
product  may  cancel  each  other,  and  that  the  number  of  terms  in  a 
product  of  polynomials  may  be  as  small  as  two.  Show  that  there  must  be 
at  least  two  terms  in  such  a  product  (cf.  Ex.  24). 

26.  When  both  multiplier  and  multiplicand  are  arranged  according  to 
the  powers  of  some  letter,  the  actual  work  of  multiplying  may  be  some- 
what shortened,  thus  : 

Multiply  3a;4-2a;3_5x2  +  (Ja:-4  by  7a:2  — 3a;  +  2. 

Ordinary  Process  Shorter  Process 

3  X*—  2  a;8-  5  x^+  6  a;  -  4  3  x^—  2  x^-  5  x^+  6  a;  —  4 

7  a;2_  3  a;  +  2  7  a;^—  3  a;  +  2 


21a;6- 

-14  x^—'So  a;4+42  a;3- 

-28  .i;2 

21x6- 

-14  a:5-35  a;4+42  a;8- 

-28x2 

- 

-  9  a;5+  6  x^-\-15  x^- 

-18a;2+12a; 

- 

-  9 

+  6 

+15     - 

-18 

+12  a; 

+  6  a;4-  4  a-3- 

-l()a;2+12a;- 

-8 

•+  6 

-  4     - 

-10 

+13    - 

-8 

21  a;6— 23  x6-23  a;4+53  a:3-5(i  x2+24  a;— 8     21  x6-23  a;5-23  x^+ss  a;8-56  x2-(-24  x— 8 
Perform  Exs.  5-9  by  this  shorter  process,  and  check  the  work. 

27.  Since  the  powers  of  the  letter  of  arrangement  in  the  multiplication 
in  Ex.  26  follow  one  another  in  regular  order,  in  each  partial  product, 
the  process  may  be  still  further  abridged  by  omitting  the  letters  until  the 
very  end.     This  is  known  as  the  method  of  detached  coefBlcients. 

Thus,  to  multiply  3  x*  -  2  x^  —  5  x2  +  6  x  —  4  by  7  x2  -  3  x  +  2,  write  only  the 

coefficients :  3 2 5+6 4 

7-   3+    2 

21  —  14  —  35  +  42  —  28 
-  9+  6  +  15-18  +  12 

+   6-   4-10  +  12-8 
21  —  23  —  23  +  53-56  +  24  —  8 
i.e.,  the  product  is    21  xS  —  23  xS  —  23  x^  +  53  x3  —  56  x2  +  24  x  —  8. 
Perform  Exs.  5-9  by  the  method  of  detached  coefficients. 


62  ELEMENTARY  ALGEBRA  [Ch.  V 

28.  Since,  for  example,  7325=  7(10)3+ 3(10)2  +  2  (10)  +  5,  jg  not 
ordinary  arithmetical  multiplication  performed  by  means  of  detached 
coefficients?    Only  the  coefficients  of  the  various  powers  of  10  are  used. 

29.  Any  absent  term,  in  the  regular  order  of  arrangement  of  a 
polynomial  to  be  multiplied  by  using  detached  coefficients,  should  be 
inserted,  with  zero  for  its  coefficient. 

Thus,  multiply  3  a;4  -  2  xs  +  6x-i,  i.e.,  dx^  -  2x^  +  Ox^+ 6x  -  i,hj 
bx-2. 

Compare  this  with  such  multiplications  in  arithmetic  (see  Ex.  28). 

30.  Multiply  2  a^  —  5  a  +  1  by  4  a  —  2,  using  detached  coefficients. 

31.  Multiply  6  x*  —  2  a;2  —  5  by  3  x^  +  5  a;,  using  detached  coefficients. 

II.    DIVISION 

43.  Law  of  exponents  in  division.  Assuming  for  the  present,  as 
in  arithmetic,  that  the  quotient  is  not  changed  if  equal  factors  be 
cancelled  from  dividend  and  divisor,  the  law  of  exponents  in 
division  is  easily  discovered. 

For  example,  —^  =  — — '- — '- — '—       [Definition  of  exponent 

i.e.f  a'^-i-a^  =  a^'^. 

Similarly,  x^ -h  x^  =  x^~'^  =  oc^ ; 

and  s^  _i.  g8  _  —  _  — 

In  precisely  the  same  way,  it  follows  that  if  m  and  n  are  any 
two  positive  integers,  then 

a"*  -f-  a"  =  a"*"",  when  m  >  ri,*   . 

a*"  -J-  a"*  =  1      ,  when  m  =  n, 

and  a"*  -7-  a"  = ,  when  in  <  n. 


*  The  symbols  >  and  <  stand,  respectively,  for  "  is  greater  than  "  and  "  is  less 
than  "  ;  thus,  m  >  n  is  read :  "  m  is  greater  than  n." 


42-44]  DIVISION  63 

44.  Zero  and  negative  exponents  defined.  Thus  far  the  symbol  a" 
has  been  defined  only  when  n  is  a  positive  integer ;  we  are  there- 
fore still  free  to  say  what  we  shall  mean  by  such  symbols  as  a~^ 
and  a".  It  will  be  found  advantageous  to  agree  that,  when  such 
symbols  present  themselves  in  any  operation,  a"  shall  be  inter- 

1  * 
preted  to  mean  1,  and  a~*  shall  mean  — • 

Under  this  definition  of  a"  and  a~*,  the  three  expressions  for  the 
quotient  of  a*"  -r-  a",  which  are  given  in  §  43,  may  be  replaced  by 
the  single  expression        ^m  _^  ^»  _  ^m-n 

whether  m  ">  n,  m  =  n,  ov  m  <  n. 

For,  when  m  =  n,  then  a^'-i-a''  =  a"*"",  because  then  a"*  h-  a^  is 
manifestly  1,  and  a™""  is  a",  which  is  also  1.    Again,  when  m<in, 

thena'"-T-a"= (§  43),  but  by  the  above  definition =  a-^"-"*) 

=  a'^"'*,  so  that  even  in  this  case  a'^-^-a''  =  a"'~". 

Hence,  with  this  extended  meaning  of  an  exponent,  the  quo- 
tient of  any  two  powers  of  a  given  numher  is  that  power 
of  the  number  whose  exponent  is  the  exponent  of  the  divi- 
dend minus  that  of  the  divisor. 

EXERCISES 

1.  What  is  the  meaning  of  x^'i  oi  x^l  . 

2.  How  many  x's  in  the  product  of  x"^  by  a;^?  How,  then,  may  this 
product  be  most  simply  written  ? 

3.  How  is  the  exponent  of  the  product  of  two  or  more  powers  of  any 
given  number  obtained?     Why? 

4.  Since  x"^  •  x^  =  x'^^,  what  is  the  quotient  when  x^^  is  divided  by  x^  ? 
Why  ?     What  is  the  quotient  of  x^'^  divided  by  x'^  ? 

5.  What  is  the  quotient  of  N^  divided  by  N^"^  of  y^^  divided  by  ^/5? 
of  p^^  by  p'  ?  of  x^  by  x*"  ? 

6.  How  is  the  exponent  of  the  quotient  of  two  powers  of  any  given 
number  obtained  ?    Why  ? 

*  In  extending  the  meaning  of  any  symbol  already  in  use,  there  is  one  principle 
that  should  always  be  observed,  viz.,  the  extended  meaning  should  be  such  that 
any  rules  of  operation  already  established  for  the  symbol  in  question  shall  not 
be  disturbed  (cf.  Ex.  9,  below). 


64  ELEMENTARY  ALGEBRA  [Ch.  V 

7.  With  exponents  restricted  to  positive  integers,  could  one  say  that 
T"  -f-  a;''  =  a;""'"  without  knowing  the  relative  values  of  n  and  r? 

8.  What  meaning  is  it  necessary  to  give  to  zero  and  negative  exponents 
so  that  x"  -f-  a;''  may  equal  x"-'",  even  when  n  =  r  and  when  rKir'i     Why  V 

9.  In  §  37  it  is  shown  that  a"*  •  «"=:a"»+"  when  a  represents  any  num- 
ber whatever,  and  m  and  n  are  any  two  positive  integers ;  show  that  this 
equation  is  still  true  if  m,  or  n,  or  both  m  and  n,  have  zero  or  negative 
values  (cf.  footnote,  p.  63). 

10.  What  is  the  meaning  of  m-^'i  oi  x^l  of  (-)~^?  Is  a"  equal  to 
x^  even  when  a  is  not  equal  to  a;  ?     Why  ? 

11.  What  is  the  product  of  x^  by  x-^1  Is  it  a^+i-^)  ?  Why?  What 
is  the  quotient  of  a^  divided  by  a^?  Is  it  a^~^?  Why?  What  is  the 
quotient  of  N^  -  iV-2?     Is  it  iV5-(-2)  ?     Why  ? 

45.  Division  of  monomials.  Since  division  is  the  inverse  of 
multiplication,  i.e.,  since  the  quotient  multiplied  by  the  divisor 
equals  the  dividend  [§  3  (iv),  note  1],  therefore  it  follows  from  the 
method  of  multiplying  monomials  (§  38)  that  the  coefficient  of 
the  quotient  of  two  monomials  is  tl%e  coefficient  of  the  divi- 
dend divided  hy  that  of  the  divisor,  and  the  exponent  of 
every  letter  in  the  quotient  is  the  exponent  of  that  letter  in 
the  dividend  diminished  hy  its  exponent  in  the  divisor. 

E.g.,  12  a5x8  ^  4  a'^x^  =  ^^a^-'h:,^-^,  i.e.,  3  aH^ ;  - 18  a^¥ -^Q  a«b^  =  Ili^a4-863'-2 

=  -  3  a65 ;  and  5  m^x^  -f- 10  m'^x^=  j%  m^-^x^-^  =  i  m^x-^  =  1  ?^^  (§  44) . 

Dividing  one  monomial  by  another  may  also  be  accomplished  by  cancellation, 
as  in  §  43.  To  test  the  correctness  of  a  quotient,  multiply  it  by  the  divisor ;  the 
product  should  be  the  dividend. 


s 


EXERCISES 


1.  What  is  the  quotient  of  Q  a^  divided  by  2  a?  of  15 a^'^  divided 
by  3  a2a-4?  of  12  m^x^  divided  by  -  4  a:2? 

2.  How  is  the  sign  of  a  quotient  determined?  the  coefficient?  the 
letters  ?   their  exponents  ? 

3.  How  may  the  correctness  of  a  quotient  be  tested  ?  Perform  the 
following  indicated  divisions,  and  test  the  result  in  each  case : 
18  a^x^  -f-  3  ax2;  15  hY  -  (-  6  Jip^)  ;  and  -  |  mh^  divided  by  -  |  mh\ 


44-46]  DIVISION  65 

4.  What  is  the  sign  of  the  product  of  two  monomials  each  of  which 
is  positive  V     Of  their  quotient  ? 

5.  Answer  the  same  questions  as  in  Ex.  4  if  each  of  the  monomials  is 
negative ;  also  if  one  is  positive  and  the  other  negative. 

6.  If  two  monomials  have  like  signs,  what  is  the  sign  of  their  product  ? 
of  their  quotient?  In  multiplication  how  is  the  exponent  of  any  particu- 
lar letter  in  the  product  obtained?  in  division? 

7.  Multiply  5  a^b^  by  2  a^c^ ;  3^  ?n%3  by  2  vix^t/  ;  2^a^x-^  by  -  6  a^xh  ; 
and  -  -m-'Y  by  j%  ah-'^k^. 

8.  Divide  18  mV  by  -  3  wV^ ;  -  -a^x-^  by  |f  a-^x^ ;  and  (D^n^z^  by 

3^ 
(—  j^^)%22.     Also  test  the  correctness  of  the  result  in  each  case. 

9.  Divide  \  h^kH-^  by  -  |  h^k'^]  -  27  ahn-^x'^  by  -  ^  iif-xy^\  2  x^+^ 
by  6a;"»;  and  15  a%^^"  by  ^axP+^y"".  Also  test  the  correctness  of  the 
results. 

10.  Show  that  even  when  some  of  the  exponents  are  negative,  as  in 
Exs.  8  and  9,  the  exponent  of  any  letter  in  the  quotient,  of  one  monomial 
divided  by  another,  is  the  exponent  of  that  letter  in  the  dividend  dimin- 
ished by  its  exponent  in  the  divisor. 

11.  Based  upon  the  definition  of  such  a  symbol  as  x-%  given  in  §  44, 

show  that  x^y-^  =  —  :  that  6a%-3^-4  =  -— ;   that  - — n^^T^-,;  and 

, ,     ,   m^n^x~'^  _  a~^rrfin^ 
a*x^y^  x^y^ 

12.  Following  the  suggestion  of  Ex.  11,  show  that  a  factor  may  be 
transferred  from  the  numerator  to  the  denominator  of  a  fraction,  and 
vice  versa,  by  merely  changing  the  sign  of  its  exponent. 

46.  Division  of  a  polynomial  by  a  monomial.  Since  the  quotient 
multiplied  by  the  divisor  always  equals  the  dividend  [§  3  (iv)], 
therefore  the  quotient  of  a  polynomial  divided  by  a  monomial 
must,  by  §  39,  be  a  jjolynomial  whose  separate  terms  being  multi- 
plied by  the  divisor  produce  the  separate  terms  of  the  dividend ; 
hence  this  quotient  is  obtained  by  dividing  each  term  of  the  divi- 
dend by  the  divisor. 

E.g.,  (15  a2x8  _  lo  hx^y  +  c2a;2)  -j. 5  a:2  =  3  a2x  —  2  hx'^y  +  \  c^. 


66  ELEMENTARY  ALGEHRA  [Oh.  V 

EXERCISES 

1.  What  is  meant  by  saying  tliat  division  is  the  inverse  of  multi- 
plication V 

2.  Since  (^a  -\-h  —  c  +  d)  •  s  =  as  +  bs  -  cs  +  ds,  what  must  be  the  quo- 
tient of  (as  4-  bs  —  cs  +  ds)  divided  by  s  ?    Why  ? 

3.  What  is  the  quotient  of  15  ax^  —  6  a*bx  -f  21  a^x^y"^  divided  by  3  ax  ? 
Why? 

4.  How  may  any  pohmomial  whatever  be  divided  by  a  monomial? 
How  are  the  signs  of  the  several  quotient  terms  determined?  their 
coefficients  ?   their  letters  ?  their  exponents  ? 

5.  Divide  6  a^x^  -  9  ab^x^  -  15  a^c^x^  by  3  ax^ ;  also  by  -  3  ax^. 

6.  Divide  —  x  +  4  ax^  —  3  m^x  —  6  ainx  by  —  ar ;  also  by  2  x. 

7.  Divide  — m  —  n  +  a:  —  aby—  1. 

8.  Divide  26  a^m^  -  52  a%m^  -  39  a^m^x^  by  - 13  a%2;  also  by  13  a%2. 

9.  Divide  -  10  r^s^y^  -  25  k^rh^  +  15  ad^r^s^  by  5  rh ;  also  by  -  5  r%2. 

10.  Divide  iam^-Q  a'^x^  +  3  a-^mx  by  |  a2:c-i. 

11.  What  is  the   meaning  of  a  negative  exponent?  of  a  zero  expo- 
nent?    How  may  the  correctness  of  an  exercise  in  division  be  verified? 

Perform  the  following  indicated  divisions  and  verify  the  results : 
/    12.    ( -  a2m3  -  4  a%^x-^  -f-  6  a-'^WmH -^)^(-\  aWcH'^). 
[What  is  the  effect  of  such  a  factor  as  a"  in  any  tgnn  ?] 

13.  (-1  m2x-2  +  \  chn'^x'^  -  \  a^mr'^x'^)  -f- 1  a^rrrH^. 

14.  {x{x  4-  yy  -  x\x  +  2/)8  -f  a;3(a:  -f  2/)2}  -  {-  x{x  +  2/)2}. 

15.  {_  (a  _  6)  -  2(a  -  by  +  3(a  -  &)4}  -^  {-  (a  -  6)}. 

16.  (a™  -  2  a'^+i  -  5  a^+2  +  3  a"*-!) -=- 1  a"*. 

17.  (2«+4-3  2"-i  +  4a22)^(_  ign-i). 

18.  (a"&"  -  I  a"-i6''+i  +  ^^  a^-^ft^+s)  -^  f  anj-n 

47.   Division  of  a  polynomial  by  a  polynomial.     Since  (see  §  42) 

(4a^+3a;-2)  •  (a^-2a;2+7a;4-5)=4a^-5 0^4+20 a^4-45a;2+a;-10, 

therefore,  with  this  last  expression  as  dividend,  and  x'—2qi?-\-1x-\-o 
as  divisor,  the  quotient  must  be  4  ar^  +  3 .'«  —  2,  i.e., 

(4a;«-5a;*-f20arV45aj2+a;-10)-i-(a:3-2aj24-7a;+5)=4a^4-3a;-2. 

The  process  of  obtaining  this  quotient  from  the  given  dividend 
and  divisor  will  now  be  explained. 


46-47]  DIVISION  67 

Since  the  dividend  is  the  product  of  the  divisor  by  the  quotient, 
therefore  the  highest  term  in  the  dividend  is  the  product  of 
the  highest  term  in  the  divisor  multiplied  hy  the  highest 
term  in  the  quotient,  and  therefore,  if  4  ar^,  the  highest  term  in 
the  dividend,  be  divided  by  y?,  the  highest  term  in  the  divisor,  the 
result,  4  a^,  is  the  highest  term  in  the  quotient. 

Moreover,  since  the  dividend  is  the  algebraic  sum  of  the  several 
products  obtained  by  multiplving  the  divisor  by  each  term  of  the 
quotient,  therefore,  if  4  ar*  —  8  a;*  +  28  a^^  +  20  cc^,  the  product  of  the 
divisor  by  the  highestterm  of  the  quotient,  be  subtracted  from 
the  divrdenci,  the  remainder,  viz.,  3  a;"*  —  8  a^  4-  25  oi?  +  x  — 10,  is  the 
sum  of  the  products  of  the  divisor  multiplied  by  each  of  the  other 
terms  of  the  quotient  except  this  one. 

Eor  the  same  reason  as  that  given  above,  if  3  x^,  the  highest 
term  of  this  remainder,  be  divided  by  a^,  the  highest  term  of  the 
divisor,  the  result,  3  x,  is  the  next  highest  term  of  the  quotient. 

By  continuing  this  process  all  of  the  terms  of  the  quotient  may 
be  found.     It  is  convenient  to  arrange  the  work  as  follows : 


a;3- 

=.2ic2d 

-7a;i5 

4.5 

S1-3X 

-2 

^ 

4a;5-5a;4  +  20x3  +  45a;2+x-10 
(,^;3-2j-2  +  7x  +  5)-4x2=     4x5-8x4-1-28x3  +  20x2 

V        3»4^  8x3  +  25x2  +  x-10 
(x3-2x2+7x  +  5)  .3x  =  3x4-  6x3  +  21x2+ 15 x  Quotient 

-  2x8+  4x2  — 14a;  — 10 
(x3-2x2  +  7x  +  5)- (-2)=  -  2x3+  4x2-14-x-10 

0 
Check 
When  x  =  l,  dividend  =  55,  divisor  =11,  and  quotient  =  5,  as  it  should. 

Even  if  it  is  not  known  beforehand  that  the  dividend  was 
actually  obtained  as  the  product  of  two  polynomials,  the  process 
of  division  may  still  be  applied  as  above. 

The  method  just  now  explained,  which  may  be  employed  to 
solve  any  example  whatever  of  this  kind,  may  be  formulated 
into  the  following  rule : 

(1)  Arrange  both  dividend  and  divisor  according  to  the 
descending  {or  ascending)  powers  of  some  one  of  the  letters 


\ 


68  ELEMENTABY  ALGEBRA  [Ch.  V 

involved  in  each,  *  and  write  the  divisor  at  the  right  of  the 
dividend. 

(2)  Divide  the  first  term  of  tlxe  dividend  hy  the  first  term 
of  the  divisor,  and  write  the  result  as  the  first  term  of  the 
quotient. 

(3)  Multiply  the  entire  divisor  by  this  first  quotient  term, 
and  subtract  the  result  from  the  dividend. 

(4)  Treat  this  remainder  as  a  new  dividend,  arranging 
as  before,  and  repeat  this  process  until  a  zero  remainder  is 
reached,  or  until  the  remainder  is  of  lower  degree  in  the 
letter  of  arrangement  than  the  divisor. 

Note.  Since  each  remainder  is  of  lower  degree  in  the  letter  of  arrangement, 
than  the  preceding  one,  therefore  it  is  always  possible  to  comply  with  (4)  in  the 
rule  just  given.  If  a  zero  remainder  is  reached,  then  the  division  is  said  to  be 
exact ;  otherwise  the  complete  quotient  consists  of  an  entire  algebraic  expression 
plus  a  fraction  whose  numerator  is  the  last  remainder  and  whose  denominator  is 
the  given  divisor. 

EXERCISES 

Divide  (and  check  your  results)  ; 
1.   a:2  +  7  a:  +  12  by  x  +  3.  2.   a;2  -  a:  -  20  by  a;  -  5. 

3.  &2  _  6  6  _  16  by  &  +  2. 

4.  p^  +  ^p^+Qp'^-\-bp-\-2  byp2  +  j9  +  i. 

5.  2  a;4  +  6  x2  -  4  a:  -  5  a:3  +  1  by  a;2  -  a:  +  1. 

6.  3  a*  +  3  a2  +  3  +  3  a  +  a6  +  5  a8  by  1  +  a-t 

7.  4  a:?/2  +  8  x8  +  2/8  _^  8  x'^y  by  y  +  2  x.* 

8.  6  aH"^  -  4  a^a;  -  4  ax^  +  a*  +  a;*  by  a^  +  a;2  -  2  ax. 

9.  2a^-\-B  -oa^k-4.  ak^  +  6  a'^k^  by  k^  +  a^  -  ak. 

10.  If  the  quotient  be  multiplied  by  the  divisor,  how'  must  the  result 
compare  with  the  dividend?  What  must  the  result  be  if  the  dividend 
be  divided  by  the  quotient  ? 

*  If  there  is  more  than  one  letter  involved  in  the  given  polynomials,  then  the 
expression  "  highest  term  "  in  the  explanation  on  p.  67  is  to  be  replaced  by  "  term 
of  highest  degree  in  the  letter  of  arrangement." 

t  Just  as  in  "long  division"  in  arithmetic,  so  here,  some  labor  may  be 
saved  by  bringing  down  only  so  much  of  the  remainder  at  any  stage  of  the 
work  as  is  needed  in  the  next  step. 


47]  DIVISION  69 

11.  If  the  partial  quotient,  at  any  stage  of  the  process  of  division,  be 
multiplied  by  the  divisor,  and  the  coiTesponding  remainder  added,  how 
must  the  result  compare  with  the  dividend? 

12.  Could  the  principles  involved  in  Exs.  10  and  11  be  employed  as 
a  check  upon  the  correctness  of  an  exercise  in  division?  Is  this  check 
more  or  less  conclusive  than  that  given  in  connection  with  the  solution  on 
p.  67?    Why? 

13.  Is  it  necessary  or  merely  convenient  to  arrange  both  dividend  and 
divisor  according  to  the  descending  or  the  ascending  powers  of  some 
letter  contained  in  each?  Could  not  the  highest  term  of  the  dividend 
be  divided  by  the  highest  term  of  the  divisor  in  whatever  order  the 
terms  of  these  expressions  are  written? 

14.  Divide  2  a:^  +  a;*  +  49  a;2  -  13  a;  -  12  by  x^  -  2  a;2  +  7  a;  +  3.* 

Divide  (and  check  the  results)  : 

15.  v^  -  r*  -  1  +  2  y  +  y3  _  y2  by  ^  -  I  -\- v^. 

16.  a5  -  41  a  -  120  by  a^  +  4  a  +  5. 

17.  m4  +  16  +  4  m2  by  2m  +  m^-{-  4. 

18.  cd  -  d^  -{-  2  c^  hj  c  -{-  d.  19.    x^  -  if  hy  x  -  y. 
20.   a-*  -  16  64  by  a  -  2  h.  21.   h^  -  F  by  h^  +  k\ 

22.  a^"'  -  a;2™  by  a"  -  rr".  23.   m2«  +  11  w«  +  30  by  m»  +  6. 

24.  x'^+^y"  —  4  a:w+«-i^2n  _  27  x'^+w-s^/S"  +  42  x'^+'^-^y'^  by  a,""  +  3  x'^-hj'^ 
-  6  x-^-hf^. 

25.  ^^x^-lx^y-\-\%x''y''^r\xy^hy  \x^\y. 

26.  1.2  ax^  -  5.494  a^x^  +  4.8  aH'^  +  0.4  a^a:  -  .478  a^  by  6  aa:  -  2  a^. 

27.  (3  x*  -  1  +  3  a:  +  6  a:2  +  7  a;3)  (1  +  a;2  -  a:)  by  x  +  1  +  a:^. 

28.  a5  -  &5  by  (a^  +  W)  (a  +  &)  +  a^V^. 

29.  10  xV  _|_  3.5  _  10  x2?/8  +  5  x?/*  -  5  x^!/  -  ?/5  by  x^  +  y2  _  2  xy, 

30.  2  x2  -  2  ?/2  _  3  ;22  _  3  a.^/  _  5  2.2  _  52/s  by  x  -  2  3/  -  3  z. 

31.  x4  -  3  x3  +x2  +  2  X  -  1  by  x^  -  x  -  2. 
[In  Ex.  31  the  complete  quotient  is 

x2  —  2  X  + 1  +    ~^  ;  compare  $  47,  note.] 

32.  x8  +  X  -  25  by  X  -  3.  33.   a^  -  1  by  a  +  1. 

34.   2  s8  -  3  s  +  8  by  s2  _  4. 


*  Since  there  is  no  term  in  x^  in  the  dividend,  care  must  be  used  to  keep  the 
remainders  properly  arranged  (of.  Ex.  29,  p.  62). 


70  ELEMENTARY  ALGEBRA  [Ch.  V 

35.  4  m^x^  -  8  m^x^  +  40  m^^  +95  by  5  +  3  7nx  -  m^x^ 

36.  abc  +  ax'^  +  a;'  +  ahx  +  feo^^  -)_  c:r2  _|_  ^(,^  _|.  j^^  ^y  ^2  ^  ^j^  ^  ^j^  ^  ^^_ 

Since  a;  occurs  in  more  terms  than  any  other  letter,  it  will  be  best  to  arrange 
the  work  in  Ex.  36  thus : 

x2+(a  +  &)a;  +  a6 


x  +  c 


a:8  +  (a  +  6  +  c)a;2  +  (ab  +  ac  +  bc)x  +  abc 
xs+{a  +  b)x^  +  abx 

cx^  +  {ac  +  bc)x-{-  abc 

cx^  +  (ac  +  bc)x  +  abc 
0 

37.  adx*-\-cf-^bfx-\-bex^  +  ecx  +  bdx^+(af+cd)x^-\-aex^  by  ax^+Jx  +  c. 

38.  ay-^  —  aby  +  y^  —  by^  —  acy  —  cy^  -\-  bey  +  abc  by  y^  —  ab  —  by  +  ay. 

39.  14:xy^+6x^-4:y^-lQx^y-2x'^-2y^  +  4xy  by  3  a;  -  1  -  2  3/. 

40.  7  a;3  +  xs  +  2  x4  -  46  a;  +  6  a:2  -  120  by  4  a;  +  5  +  a;2. 

41.  7  a2  _  6  a3  +  a4  -  4  a  -  12  by  3  -  2  a  +  a2. 

42.  (4  m4  -  5  m262  +  ^4)  (5  ^2^  +  ^^s  +  ^3  _^  5  ^.^2) 

by  (2  w2  -  3  w6  +  ^2)  (a;2  +  ^2  _^  4  3:3/). 

43.  a^  -  63  _,.  c3  +  3  a&c  by  a^  +  b'^  +  c^  +  ab  -  ac  +  be. 

44.  a:6  -  6  a:  +  5  by  x2  -  2  a;  +  1. 

45.  Divide  Sab  ■{■  a^ -\-b^  by  a  +  6,  arranging  according  to  descending 
powers  of  a.  Perform  this  division  also  with  the  expressions  arranged 
according  to  descending  powers  of  b,  and  compare  the  two  results. 

46.  Divide  2  xy^  +  3  x*  —  4  a:2^2  _  7  x^y  +  y^  by  x^  +  y'^  —  xy,  arranging 
first  according  to  powers  of  x,.then  according  to  powers  of  y,  and  com- 
pare the  results. 

47.  As  has  just  been  seen,  in  Exs.  45  and  46,  the  form  of  the  quotient 
depends  upon  the  choice  of  the  letter  of  arrangement  lohen  the  division  is 
not  exact;  prove  that  this  is  not  the  case  when  the  division  is  exact. 

48.  Divide  p^ -\-  q^  by  p  -\-  q,  until  4  quotient  terms  are  obtained. 

49.  Divide  a  by  a  —  ar,  to  5  quotient  terms. 

50.  Divide  1  by  1  —  r,  to  8  quotient  terms. 

51.  Divide  1  by  1  —  mx,  to  4  quotient  terms. 

52.  Divide  a:«  —  y"  by  x  +  y,  to  8  quotient  terms.  What  does  this 
quotient  become  when  w  =  2,  3,  4,  •.-?  What  is  the  remainder  when 
n  =  2,  4,  6,  8,  ...?  when  n  =  3,  5,  7,  -  ? 

53.  Examine  the  quotient  (x»  —  ?/")  -^  (x  —  y)  under  the  same  circum- 
stances as  in  Ex.  52.    Also  (a"  +  6")  ^(a+ft),  and  (p"  +  5")-^(jo-^)- 


iM8]  DIVISION  71 

/       54.   Some  labor  may  often  be  saved  in  an  exercise  in  division  by  using 
/   the  coefficients  only,  and  omitting  the  letters  until  the  end. 

/         Thus,    (4a:5-5a;4  +  20cc3  +  45a.2-}-a._io)-4-(a;8-2a;2  4-7a;  +  5),  with  letters 
I      omitted,  becomes 

4-5  +  20  +  45  +  1-10 

4  —  8  +  28  +  20 


3- 

8  +  25+  1 

3- 

6  +  21  +  15 

— 

2+   4-14- 

-10 

- 

2+   4-14- 

-10 

1-2+7+5 


4  +  3-2,  i.e.,  4a;2  +  3a;-2. 


This  example  has  already  been  solved  on  p.  67 ;  the  student  should 
carefully  compare  the  two  methods.  He  should  also  note  that  this  last 
method  —  called  the  method  of  detached  coefficients  —  is  altogether 
similar  to  "long  division"  in  arithmetic,  and  analogous  to  that  em- 
ployed in  Ex.  27,  p.  61. 

\By  the  method  of  detached  coefficients,  perform  Exs.  1,  4,  5,  6,  8, 
nd  9. 

55.  In  using  the  method  of  detached  coefficients,  if  any  powers  of 
the  letter  of  arrangement  are  absent  they  must  be  supplied,  giving  them 
zero  coefficients ;  compare  this  with  Ex.  29,  p.  62.  Solve  Ex.  14  by  this 
method,  writing  the  dividend  thus :  2  x^  +  a;*  +  0  a;^  +  49  a:^  —  13  a;  —  12. 

56.  Solve  Exs.  16  and  17,  using  detached  coefficients. 

57.  Divide  a;^  +  4  x^  —  7  x  +  2  by  ar  —  a,  and  show  that  the  remainder 
is  what  would  he  obtained  by  substituting  a  for  x  in  the  dividend. 

58.  Divide  5  m^  —  8  m  +  3  by  m—  r,  and  compare  the  remainder  with 
the  dividend.  Similarly,  divide  z^  —  Sz^-^z^—1  hj  z  —  b',  y^— 3y+l 
by  v-2;  and  2  x*  +  5  a;^  -  a;  +  10  by  ar  -  c. 

48.  Remainder  theorem.  In  Ex.  57  on  this  page  it  is  shown 
that  when  a^  +  4a^  —  7a;  +  2is  divided  by  x—a  the  remainder  is 
a^  +  4a^  —  7a  +  2;  i.e.,  the  remainder  is  what  would  be  obtained 
-^by  substituting  a  for  x  in  the  dividend. 

To  show  that  this  relation  between  dividend  and  remainder  is 
not  accidental,  but  that  it  is  always  true  when  a  polynomial  in 

X  is  divided  by  x  —  a,  let  Ax""  +  Bx''-'^  +  Cx""-^  + \- Hx -\- K 

represent  any  such  polynomial  whatever,  arranged  according  to 
descending  powers  of  x,  and  let  Q  and  E,  respectively,  represent 


72  ELEMENTARY  ALGEBRA  [Ch.  V 

the  quotient  and  remainder  when  this  polynomial  is  divided  by 
X  —  a;  then,  since  the  dividend  equals  the  quotient  times  the 
divisor,  plus  the  remainder, 

Ax''  +  Bx^-'  +  Caj"-2  -\-...+Hx  +  K=Q{x  -  d)  +  B. 

Moreover,  since  the  second  member  of  this  equation,  when  mul- 
tiplied out,  must  be  exactly  like  the  first  member,  therefore  this 
equation  is  true  for  all  values  that  may  be  assigned  to  x ;  but  if 
the  value  a  be  given  to  x,  the  equation  becomes 

Aa^  +  Ba^-'  +  Ca"-^  +  ...  +  Ha  +  K=  R* 

hence,  in  every  such  division,  the  remainder  may  be  immediately 
written  down  by  substituting  a  for  x  in  the  dividend. 
It  also  follows  from  this  theorem  that  if 

then  Ax""  +  Bx""-^  +  Cx""-^  -\ +  Hx  +  K  is  exactly  divisible  by 

x  —  a,  for  in  that  case  the  remainder  is  zero ;  and  conversely. 

EXERCISES 

1.  What  is  the  remainder  when  3  a:*  —  2  a:  +  1  is  divided  by  a:  —  c  ?  by 
x  —  a'i  byx  —  2?    Answer  these  questions  by  means  of  §  48. 

2.  What  is  the  remainder  when  y^-\-2  y^—\i:y  —  ^  is  divided  by  ?/— a? 
by  2/  —  A;?  by  2/  +  2,  i.e.,  by  y  —  (—  2)  ?  by  ?/  —  3  ?     Try  the  last  two  cases. 

3.  Is  ar  —  3  an  exact  divisor  of  x*  —  4  x^  +  5  a:  +  12  ?  Answer  without 
actually  performing  the  division. 

REVIEW  QUESTIONS-CHAPTERS  l-V 

1.  Define  the  following  operations :  addition ;  subtraction  ;  multipli- 
cation ;  division.     Which  of  these  are  inverse  operations  ?     Explain. 

2.  Point  out  at  least  one  advantage  which  the  definition  of  multipli- 
cation as  given  in  §  3  (iii)  has  over  the  usual  arithmetical  definition. 

3.  In  a  number  system  consisting  of  positive  integers  only,  is  division 
always  a  possible  operation  ?  How  must  this  number  system  be  enlarged 
so  that  division  may  be  always  possible  ? 

Answer  these  questions  with  regard  to  subtraction  also. 

*  Since,  in  that  case,  Q{z  —  a)  becomes  Q{a  —  a),  i.e.,  0. 


48]  DIVISION  73 

4.  Point  out  at  least  two  advantages  of  using  letters  to  represent 
numbers. 

5.  Define  and  illustrate  a  negative  number.  How  may  a  negative 
number  be  subtracted  from  any  given  number?  State  and  prove  the 
"  law  of  signs  "  for  multiplication  of  negative  numbers.    Also  for  division. 

6.  How  may  a  parenthesis  which  incloses  several  terms,  and  which 
is  preceded  by  the  minus  sign,  be  removed  without  affecting  the  value 
of  the  expression^    Why  ? 

7.  Define  an  algebraic  expression;  a  term;  a  binomial;  a  poly- 
nomial; a  coefficient;  an  exponent;  the  degree  of  a  term,  and  of  an 
integral  polynomial. 

8.  State  the  several  steps  in  solving  an  algebraic  problem.  What 
axioms  are  frequently  used  in  such  solutions?  What  is  meant  by 
"  checking  the  work  "  ? 

9.  How  are  two  or  more  similar  monomials  added?  State  a  rule  for 
subtracting  one  polynomial  from  another. 

10.  Prove  that  a"»  •  a"  •  qp  =  0"*+"+^  if  a  is  any  number  whatever  and 
m,  n,  and  p  are  positive  integers. 

11.  How  may  the  product  of  two  or  more  monomials  be  obtained  ? 

12.  Give  a  rule  for  dividing  one  polynomial  by  another.  Also  explain 
a  device  for  abbreviating  the  work.  State  two  ways  of  checking  the 
correctness  of  an  exercise  in  division. 

13.  Are  negative  numbers  ever  used  as  exponents?  Is  zero  so  em- 
ployed ?  What  is  the  interpretation  of  such  symbols  as  5-^,  a°,  and  x-«  ? 
What  is  the  advantage  of  such  exponents? 

14.  Prove  that,  under  a  proper  interpretation,  negative  and  zero 
exponents  conform  to  all  the  laws  previously  established  for  positive 
integral  exponents. 

15.  Prove  that  any  factor  may  be  transferred  from  the  numerator  of 
a  fraction  to  the  denominator,  or  vice  versa,  by  merely  reversing  the  sign 
of  its  exponent — whether  the  given  exponent  be  positive  or  negative. 


CHAPTER   VI 
COMBINATORY   PROPERTIES   OF   NUMBERS* 

49.  Introductory.  Some  combinatory  properties  of  numbers, 
the  correctness  of  which  has  thus  far  in  this  book,  and  also  in 
arithmetic,  been  assumed,  deserve  to  be  somewhat  carefully 
studied.  This  further  study  is  not  so  much  needed  to  give  the 
student  confidence  in  their  correctness  as  it  is  to  justify  the  con- 
fidence he  already  feels;  it  is  designed  to  guard  the  student 
against  drawing  conclusions  which  are  not  fully  warranted. 

To  illustrate :  since  by  actual  counting  3+5  =  8  and  5  +  3  =  8, 
therefore  3  +  5  =  5  +  3;  similarly  it  is  found  that  9+2  =  2  +  9, 
15  +  7  =  7  + 15,  etc. ;  but  merely  verifying  this  fact  in  particular 
cases  does  not  warrant  the  conclusion  that  a  +  6  =  6  +  a,  when  a 
and  b  represent  an  untried  pair  of  numbers.  So  far  as  the  above 
reasoning  is  concerned,  the  very  next  pair  of  numbers  that  is 
tried  may  prove  to  be  an  exception. 

If  a  large  number  of  verifications  could  establish  a  general  law, 
then  the  conclusion  that  a*  =  6",  for  every  pair  of  numbers,  would 
be  valid  to  one  who  had  happened  to  try  only  those  pairs  of  num- 
bers for  which  this  is  true ;  e.g.,  2^  =  4P.1f 

50.  Commutative  law  of  addition.  In  §  49  it  was  verified  that 
3  +  5  =  5+3,  9  +  2  =  2  +  9,  etc.  These  are  particular  cases  of  a 
general  principle  which  is  known  as  the  commutative  law  of  addi- 
tion. This  law  may  be  stated  thus :  the  sum  of  two  or  more 
numJbers  is  not  changed  hy  changing  the  order  in  which 
these  numbers  are  added. 

That  this  law  is  true  for  every  set  of  numbers  without  exception 
will  now  be  shown,  not  by  verifying  it  in  particular  cases,  —  that 

*  This  chapter  may,  if  the  teacher  prefers,  be  omitted  on  a  first  reading, 
t  Admitting  fractional  exponents,  which  are  introduced  later  (§  153),  the  num- 
ber of  pairs  of  numbers  for  which  a^  =  6«  is  infinitely  large. 

74 


49-50]       COMBINATORY  PBOPERTIES   OF  NUMBERS  75 

method  would  not  really  prove  anything  for  any  untried  set  of 
numbers  (§  49),  —  but  by  fundamental  considerations  based  upon 
the  primary  meaning  of  number. 

(i)  The  numbers  positive  integers.  To  show  that  a  +  6  equals 
6  +  a,  whatever  positive  integers  are  represented  by  a  and  h,  let 
there  be  a  objects  *  in  one  group  and  h  objects  in  another,  then 
a  +  6  means  the  number  of  objects  in  the  group  formed  by  adding 
the  objects  of  the  second  group  to  those  of  the  first,  and  h  -\-  a 
means  the  number  of  objects  in  the  group  formed  by  adding  the 
objects  of  the  first  group  to  those  of  the  second;  but  manifestly 
the  total  number  of  objects  in  the  two  groups  f  is  the  same 
whether  the  second  group  be  added  to  the  first  or  the  first  to  the 
second,  and  therefore        a  -{-  h  =  h  +  a. 

Similarly,  the  correctness  of  this  law  is  shown  for  any  number 
of  positive  integers. 

(ii)  The  nuwibers  negative  integers.  Since  the  sum  of  any 
number  of  negative  integers  is  found  by  getting  the  sum  of  the 
absolute  values  of  these  numbers  and  ]3refixing  to  this  sum  the 
minus  sign  (§  16),  therefore,  by  (i)  above,  the  commutative  law  is 
true  for  any  number  of  negative  integers. 

(iii)  The  numbers  integers,  some  positive  and  some  nega- 
tive. Such  a  sum  as  2  -f-(—  6)+7  is  obtained  by  first  adding  —6 
to  2  and  then  adding  7  to  that  result;  but  2  +  (-  6)  =  — 4  (§  16), 
and  —4-1-7  =  3;  i.e.,  two  of  the  negative  units  in  —  6  are  can- 
celled by  the  2,  and  the  —  4  that  remains  cancels  four  of  the 
positive  units  in  7.     Similarly  in  general. 

In  other  words,  in  adding  positive  and  negative  numbers  one 
negative  unit  cancels  one  positive  unit  and  but  one,  and  viee  versa. 
Now  neither  the  number  of  positive  units  nor  the  number  of 
negative  units  is  changed  by  changing  the  order  in  which  the 
addition  is  performed  [(i)  and  (ii)  above];  therefore  the  sum 
(the  number  of  uncancelled  units)  is  not  changed  by  changing 
the  order  in  which  the  additions  are  made. 

*  These  may  be  any  objects  whatever,  and  need  not  even  be  of  the  same  kind ; 
for  the  purpose  of  mere  counting  any  object  may  take  the  place  of  any  other. 

t  This  assumes  merely  that  an  object  may  be  removed  from  one  position  to 
another  without  destroying  its  individuality. 


76  ELEMENTARY  ALGEBRA  [Ch.  VI 

(iv)  TJie  numhers  fractions.  It  will  presently  be  shown  that 
any  given  fractions  can  always  be  reduced^to  equivalent  fractions 
having  a  common  denominator,  and  such  that  the  numerators 
and  denominators  are  integers ;  it  will  also  be  shown  (§  54)  that 

such  a  fraction  as  —   is  equal  to  m  times   -•     Assuming  this 
n  n 

for  the  present,  it  follows  that  the  commutative  law  is  true  for 

this  case  also,  for,  if  the  simplified  fractions  are  — ,  -,  -,  etc., 

n    n    n 

then  — !--  +  -  +  •••  means  m  times   -  +  »  times  -  +  q  times 
n      n      n  n  n 

_  _{_  ...  i,e.,  if  -  be  called  the  fractional  unit,  then  — h  - +  -  +  ••• 
n  n  n      n      n 

means  m  -f  p  +  g  +  •••  times  this  fractional  unit;  but,  by  (i),  (ii), 
and  (iii)  above,  the  sum  m  +  j9  +  g  +  •••  is  independent  of  the 

order  in  which  the  addition  is  performed;  therefore  — h  -  +  -  H 

n      n     n 

is  independent  of  the  order  in  which  the  fractions  are  added. 

Hence  the  commutative  law  of  addition  is  true  for  positive  and 
negative  integers  and  fractions. 

51.  Associative  law  of  addition.  Another  law  of  the  same 
general  character  as  that  given  in  §  50  above,  is  known  as  the 
associative  law  of  addition,  and  may  be  stated  thus :  the  sum  of 
three  or  more  numbers  is  not  changed  hy  grouping  together 
two  or  more  of  the  swmmands,  and  replaxAng  them  by 
their  sum. 

E.g.,  3  +  6  +  2  =  3+ (6  +  2)  =  3  +  8,  [Each  member  being  11 

and  5  +  3  +  6  +  8=5+(3  +  6) +8  =  5+9  +  8.       [Each  member  being  22 

To  show  that  this  law,  which  has  just  been  verified  in  two 
particular  cases,  is  true  for  any  set  of  numbers  whatever  (positive 
or  negative,  integers  or  fractions),  let  a,  5,  c,  and  d  represent  any 
four  such  numbers ; 

then  a+h  +  c  +  d=h  +  d-\-a-[-c  [§50 

=  (&  +  c^)  +  a  +  c  ■  [§  8 

=  a  +  (6  +  d)  +  c,  [§50 

i.e.,  the  numbers  h  and  d  may  be  grouped  together  and  replaced 
by  their  sum ;  similarly  for  any  two  or  more  of  the  summands. 


50-52]       COMBINATORY  PROPERTIES  OF  NUMBERS  77 

Observe  that  the  process  employed  in  the  proof  just  given  is  entirely  general, 
i.e.,  that  it  applies  to  any  number  of  summands  and  to  any  desired  grouping  of 
them  ;  it  consists  in  first  bringing  the  numbers  which  it  is  desired  to  group  together 
into  the  leading  places  in  the  sum  (§  50),  then  grouping  them  together  (§  8),  and 
then  putting  the  group  (which  is  a  number)  into  any  desired  place  (§  50). 

52.  Commutative  law  of  multiplication.  Another  principle 
which  the  student  has  ah-eady  used  freely,  and  which  is  of  the 
same  general  character  as  those  given  in  §§  50  and  51,  is  known 
as  the  commutative  law  of  multiplication.  This  principle  may  be 
stated  thus:  the  product  of  two  or  more  numbers  is  not 
changed  hy  changing  the  order  in  which  the  multiplica- 
tions are  performed. 

E.g.,  5  •  8  =  8  •  5.  [Each  member  is  40 

So.  too,  3.4-9  =  4-3-9  =  9-3.4,  etc.     [Each  member  is  108 

Although  the  law  which  has  just  been  stated  and  illustrated  is 
true  for  any  numbers  whatever,  its  complete  proof  necessarily 
divides  itself  into  several  parts ;  the  proof  of  its  correctness  when 
some  or  all  of  the  numbers  are  fractions  is  given  in  §  54  (iii), 
while  the  part  of  the  proof  which  concerns  integers  only  will  now 
be  given. 

(i)  Proof  for  three  positive  integers ;  also  for  two.  Let  a, 
h,  and  c  represent  any  three  positive  integers  whatever,*  and  let 
a  rectangular  array  containing  h  rows  and  c  columns  of  groups 
of  a  objects  each,  be  formed,  thus : 

c  columns 


a,  a,  a,  ••-,  a 

a^  a,  a^  '",  a 

6  rows  •  a,  a,  a,  '",  a 

[  a,  a,  a,  •••,  a 

Since  there  are  a  objects  in  each  group  and  b  groups  in  each 
column,  therefore  the  number  of  objects  in  a  column  is  a  •  6 ;  and 
since  there  are  a  •  b  objects  in  each  column  and  c  columns,  there- 

*  When  reading  this  proof  for  the  first  time,  it  may  be  best  for  the  student  to 
use  a  set  of  particular  numbers  such  as  3,  5,  and  6  instead  of  a,  b,  and  c. 


78  ELEMENTARY  ALGEBRA  [Ch.  VI 

fore  the  number  of  objects  in  the  entire  array  is  («•&)•  c,  i.e., 
a  '  b  '  c* 

Again,  the  number  of  objects  in  a  row  is  a  •  c ;  and,  since  there 
are  b  rows,  the  number  of  objects  in  the  entire  array  is  (a  •  c)  •  b, 
i.e.,  a  '  c  'b. 

But  the  number  of  objects  in  the  entire  array  is  manifestly  the 
same  when  they  are  counted  in  one  order  as  it  is  when  they  are 
counted  in  another ;  therefore 

a  -b  •  G  =  a'  c  'b,  (1) 

i.e.,  the  product  of  any  three  positive  integers  is  not  changed  by  inter- 
changing the  order  of  the  second  arid  third. 
If  a  =  1,  then  equation  (1)  becomes 

b'C  =  C'b,  (2) 

i.e.,  the  product  of  any  two  positive  integers  is  not  changed  by  inter- 
changing their  order. 

Eemark.  Since  multiplier  and  multiplicand  may  be  inter- 
changed, each  is  called  a  factor  of  the  product ;  and,  in  general, 
the  numbers  which  multiplied  together  produce  a  certain  product 
are  called  the  factors  of  that  product. 

(ii)  Proof  for  any  numher  of  positive  integers.  By  means 
of  the  proof  given  in  (i),  it  is  easily  shown  that  any  two  consecu- 
tive factors,  in  a  product  of  two  or  more  integers,  may  be  inter- 
changed without  changing  that  product. 

E.g.,  that     Jc'm-n'P'S^k-ni'p-n-s, 

may  be  shown  as  follows : 

A;  •  m  •  n  •  p  =  (^  •  m)  •  71  •  /)        .  [§  8 

=  (k-m)  'p  '71  [(i)  above 

=  k  -m  •  p  -n,  [§  8 

i.e.,  k- m-ri' p  =  k- rri' p  '  n; 

whence  k'm'n'P'S  =  k-m-p'n-s,       [Multiplying  each  member  by  s 

i.e.,  the  product  k  •  m  -  n  •  p  •  s  is  not  changed  by  interchanging  the  two  consecu- 
tive factors  n  and  p.     Similarly  in  general. 

*  The  order  of  multiplication  being  from  left  to  right  (§  8),  a  •  6  •  c  means  the 
same  as  (a  •  6)  •  c. 


52-53]       COMBINATORY  PROPERTIES   OF  NUMBERS  79 

Moreover,  by  successive  interchanges  of  two  consecutive  factors, 
all  the  factors  of  a  product  may  be  arranged  in  any  desired  order. 

Therefore,  the  product  of  any  number  of  positive  integers  is  not 
changed  by  any  change  whatever  in  the  order  of  the  factors. 

(iii)  Proof  when  some  factors  are  negative.  The  proof 
just  given  applies  also  to  products  in  which  some  of  the  factors 
are  negative,  because  the  absolute  value  of  such  a  product  is  the 
same  as  though  all  of  its  factors  were  positive ;  and  its  quality  is 
determined  by  the  number  of  its  negative  factors  (§  18,  note  1) ; 
hence  neither  the  quality  nor  the  absolute  value  of  a  product 
of  two  or  more  integers  is  changed  by  merely  changing  the  order 
of  its  factors. 

Therefore,  the  product  of  any  number  of  integers  is  not  changed 
by  any  change  whatever  in  the  order  of  the  factors. 

53.  Associative  law  of  multiplication.  As  might  be  inferred 
from  its  name  (cf.  §  51),  this  law  asserts  that  the  product  of 
any  number  of  factors  is  not  changed  hy  grouping  together 
two  or  more  of  these  factors  and  replacing  them,  hy  their 
product. 

E.g.,  2.  5 -3-7  =  2- (5-3)  •7  =  2-15.7.  [Each  member  is  210 

The  proof  *  of  this  law  is  as  follows :  the  factors  to  be  grouped 
together  may,  by  successive  applications  of  the  commutative  law 
(§  52),  be  brought  together  into  the  leading  places,  in  which 
position  they  may  be  grouped  together  and  replaced  by  their 
product  (§  8) ;  if  it  is  desired  to  group  together  still  other  factors, 
they  may  now  be  treated  in  the  same  way. 

To  illustrate  :  if  a,  b,  c,  c?,  and  e  represent  any  integers  whatever, 


then                       a  •  b  '  c  -  d  '  e  =  a  '  {b  '  e)  •  c  '  d', 

for                          a'b'C'd'e  =  b'e'a-C'd 

[§52 

=  (6  •  e)  •  a  '  c  '  d 

[§8 

=  a  '  (b  '  e)  '  G  '  d, 

[§52 

i.e.,                         a  -b  •  c  '  d  '  e  =  a'  (b  •  e)  '  c  '  d, 

which  was  to  be  shown. 

*  The  proof  is  here  limited  to  the  case  of  integers  because  it  depends  upon  §  52, 
which  is  thus  limited ;  in  §  54  (iv)  the  case  involving  fractions  will  be  considered. 


80  ELEMENTARY  ALGEBRA  [Ch.  VI 

54.  Some  fundamental  principles  involved  in  operations  with  frac- 
tions.* The  way  to  use  fractions  has  already  been  taught  in 
arithmetic,  but  the  underlying  principles  upon  which  such  use  is 
based  should  also  be  carefully  mastered  by  the  student. 

Among  these  principles  are  : 

(i)  The  product  of  two  simple  fractions^  is  a  simple 
frojction  whose  numerator  is  the  product  of  the  numera- 
tors of  the  ^iven  fractions,  and  whose  denominator  is  tJie 
product  of  their  denominators ; 

i.e.,  if  2h  q,  r,  and  s  represent  any  four  integers  whatever, 

then  P.Z^Pr. 

q      s      qs 

In  order  to  simplify  the  proof  of  (1),  let  it  first  be  observed  that: 

(a)  If  P  •n=  Q-  n,  then  P=  Q;  for  if  P  is  not  equal  to  Q,  let  P=  Q+  R 

(wherein  R  is  positive  or  negative) ,  then  P  -  n=  ( Q  +  jB)  •  n=  Q-  7i-{-  R  •  n  (^39), 
I  i.e.,  P  •  n  is  not  equal  to  Q  •  n,  which  is  contrary  to  the  hypothesis. 

(6)  It  follows  from  the  different  ways  of  counting  the  a's  in  the  rectangular 

array  in  §  52  (i)  that,  whatever  the  number  represented  by  a,  so  long  as  b  and  c 

are  integers,  ,  ,  n.      \ 

a'0'C=a'C'0  =  a-{0'C). 

(c)  To  multiply  any  number  by  the  simple  fraction  -  means  first  to  multiply 

s 
that  number  by  r,  and  then  to  divide  the  product  by  s,  for  the  fraction  -  is 
obtained  from  the  unit  in  this  way  [cf .  §  7  (v)] .  * 

p      T      pr 
The  proof  that  —  •  -  =  —  is  as  follows : 
^  q      s     qs 

p     r  p 

5"  *  i  *  ^  *  ^  =  g  •  *'  ^  ^  •  ^  •  ^  [%  (c)  above 

=  |.r.g  [§7(v) 

p 
=  —  •  g  •  r  [By  (h)  above 

=  pr; 

*  Observe  carefully  that,  in  the  following  proofs,  a  fraction  is  always  regarded 
as  an  indicated  division. 

t  By  a  "  simple  fraction  "  is  here  meant  one  whose  numerator  and  denominator 
are  integers. 


54] 


COMBINATORY  PROPERTIES  OF  NUMBERS 


81 


pr                pr 
and                    ^—  '  s  '  q  =  —  •  qs 
qs          ^      qs     ^ 

=pr; 

[By  (6)  above 
[§  7  (V) 

,                     p     r                pr 

hence            -  -  -  •  s  •  q  =  - —  s  •  a, 
q      s          ^      qs         ^' 

"Each  member  being 
_  equal  to  pr 

and  therefore          -  •  ^—^ 
q     s      qs' 

[By  (a)  above 

which  was  to  be  proved. 

(ii)  The  product  of  any  numher  of  simple  fractions  is  a 
simple  fraction  whose  numerator  is  the  product  of  the 
numerators  of  the  given  fractions,  and  whose  denominator 
is  the  product  of  their  denominators. 


For,  since 
r 


pr 


p     r 

q     s      qs 

P 

q      s      V      qs      V 

number  of  simple  fractions. 


which   is   a  simple  fraction,   therefore 


=  —  •  -  =  - —    and  similarly  for  the  product  of  any 
qs      V      qsv  j  r  j 


(iii)  The  product  of  two  or  more  simple  fra/itions  is  not 
changed  hy  changing  the  order  in  which  the  multiplicar- 
tions  are  performed  (commutative  law), 

prux^prux  [By  (ii)  above 

q    s    V   y      qsvy 

[§  52 
[By  (ii)  above 


E.g., 


purx 


qvsy 
_p    u    r    x^ 
~q    V    s   y^ 

i.e.,  the  product  of  these  fractions  remains  unchanged  by  inter- 

cl^?9f^P^^  factors  -  and  -:  similarly  in  general  for  any  num- 
s  V 

bar  of  factors,  and  for  any  desired  order. 

Note.    Since  —  is  the  same  as  m,  and  since  in  the  above  demonstrations  any 

of  the  denominators  may  be  1,  therefore  those  proofs  remain  valid  when  some  of 
the  factors  are  fractions  and  some  are  integers. 
In  particular,  it  follows  from  (iii)  that 

Inn 


1 

m  , 

i  i.e., 

,thatm.l  =  i.m  =  a 

n 

'  1  ' 

n     n            n 

82  ELEMENTARY  ALGEBRA  [Ch.  VI 

(iv)  TTie  product  of  any  numher  of  fractions  {and  inte- 
gers) is  not  changed  hy  grouping  together  any  two  or  more 
of  them  and  replacing  them  hy  their  product  (associative 
law). 

For  the  factors  to  be  grouped  may,  by  (iii)  above,  and  note, 
be  brought  together  into  the  leading  places,  in  which  position  they 
may  be  grouped  together  and  replaced  by  their  product  (§  8) ;  if 
it  is  desired  to  group  together  still  other  factors,  they  may  now 
be  treated  in  the  same  way. 

(v)  The  value  of  any  simple  frojction  is  not  changed  hy 
multiplying  hoth  numerator  and  denominator  hy  any  inte- 
ger whatever,  or  by  dividing  hoth  hy  any  integer  factor  of 
eojch. 

For,  since  tZ  =  ^,  [By  (i)  above 

(^     S        QS 

whatever  integers  are  represented  by  the  letters, 

therefore =  =^— ,  i.e.,  —  =  ±—i  [Since  -  =  1 

q    r      qr         '  q      qr  '-  r 

and,  since  this  last  equation  may  be  read  either  way,  the  proposi- 
tion is  proved. 

This  theorem  enables  one  to  reduce  fractions  to  their  "  lowest 
terms,"  and  also  to  reduce  two  or  more  given  fractions  to  equiva- 
lent fractions  having  a  "  common  denominator." 

(vi)  To  divide  hy  a  simple  fraction  gives  the  same  result 
as  to  multiply  hy  this  fraction  inverted. 

For,  let  2^  represent  any  integer  or  simple  fraction,  and  let  - 
represent  any  simple  fraction ;  then 


^ 


N^l=N^'-.(L.£\  [Since  r.?  =  ?:f=l 

8  s    \s    rj  s    r     sr 

=  iV^  r  .  r  .  !  [By  (iv)  above 

s    8    r 

=  iVT.  !,  [Since  J\r-^  .  ^=  iV,  [§  7  (v)] 

r  8    ^ 


i.e.,  N-^-=N'  -,  which  was  to  be  proved, 

s  r 


54]  COMBINATORY  PROPERTIES   OF  NUMBERS  83 

Kemark.  If  a  represents  any  number  whatever,  then  1  -i-  a  is 
called  the  reciprocal  of  a.  From  this  definition  it  follows  that  the 
reciprocal  of  a  simple  fraction  is  that  fraction  inverted;  for,  if 
-^=  1  in  the  proof  just  given,  then 

s  r     r 

(vii)  The  sum  of  two  or  more  simple  fractions  whieh 
have  the  same  denominator  is  a  fraction  whose  numera- 
tor is  the  sum  of  the  numerators  of  the  given  fractions, 
and  whose  denominator  is  the  common  denominator  of  the 
given  fractions. 

For,  let  -,  -,  and  -  represent  any  simple  fractions  having  a 
common  denominator ;  then 

-  +  -  +  -=«•  -+h  •  -  +  c  '  -  [By  (iii)  above,  note 

d     d     d  d  d  d  ljv/ 

=  (a  +  &  +  c)  .  i  [Distributive  law,*  §  39 


d 


a  +  b  -\-  c 


,  ah      G_a+h+c 

'•'•'        d^~d^d~        d 


[By  (iii)  above,  note 


I 


and  similarly  for  any  number  of  such  fractions.  If  the  given 
fractions  have  not  a  common  denominator,  they  must  be  reduced 
to  equivalent  fractions  having  a  common  denominator  [see  (v) 
above]  before  they  can  be  added. 

(viii)  Complex  fractions.  A  complex  fraction  is  usually 
understood  to  mean  a  fraction  whose  numerator  or  denominator 
or  both  are  themselves  fractions,  i.e.,  it  is  an  indicated  division  in 
which  the  dividend  and  divisor  may  themselves  be  fractions. 

*  In  §  39  it  was  proved  that  multiplication  is  distributive  as  to  addition ;  the 
student  is  advised  to  re-read  that  proof,  and  to  observe  that  the  reasoning  there 
employed  makes  no  restriction  upon  the  numbers  involved,  —  these  numbers  may 
be  integers  or  fractions,  and  positive  or  negative.  It  follows  then  that  division 
also  is  distributive  as  to  addition,  because  dividing  by  any  number  d  is  the  same 
as  multiplying  by  -  [(iii),  note,  and  (iv)]. 


84  ELEMENTABY  ALGEBRA  [On.  VI 

By  the  foregoing  principles,  and  especially  by  (vi)  above,  com- 
plex fractions  may  always  be  reduced  to  equivalent  simple  frac- 
tions, and  may  then  be  replaced  by  these  simple  fractions ;  hence 
the  commutative  and  associative  laws,  which  were  demonstrated 
above  for  integers  and  simple  fractions,  apply  to  complex  fractions 
also;  i.e.,  these  laws,  as  well  as  the  distributive  law  (§  39), 
apply  to  any  integers  and  fractions  whatever. 

55.  Zero  ;  operations  involving  zero.  Zero  may  be  defined  as  the 
result  of  subtracting  any  number  from  itself ;  it  is  represented  by 
the  symbol  0. 

E.g.^  a~a  =  0, 

whatever  the  number  represented  by  a. 

By  replacing  0  by  a  —  a  it  is  easily  shown  that 

71  +  0  =  71  =  71  —  0;  0  '  n  =  n  '  0  =  0;  and   0  -f-  n  =  0, 

where  n  represents  any  finite  number  whatever. 

Again,  since  ti -i- d  stands  for  the  number  which,  being  multi- 
plied by  d,  will  produce  n  [§  3  (iv)],  therefore  0^0  Tnay  Jiave 
any  finite  value  whatever,  because  any  finite  number  multiplied 
by  0  equals  0 ;  and  7i  -^  0  (wherein  n  is  any  finite  number)  hm 
no  finite  value  whatever,  because  no  finite  number  multiplied  by  0 
equals  ri.*  From  what  has  just  been  said,  it  is  clear  that  0  must 
not  be  used  as  a  divisor. 

EXERCISES 

1.   What  are  the  values  of  the  expression  2  n  + 1  when  n  =  1, 2, 3,  ••-,  15? 

Are  these  values  even  or  odd  ? 


m 


2.   Do  the  answers  of  Ex.  1  warrant  the  conclusion  that  2  n  +  1  rep- 
nts  an  odd  number  for  every  integer  value  of  n  (cf.  §  49)  ?    Prove 
oth  2^  +  1  and  2^  —  1  represent  odd  numbers  for  all  integer  values 


3.  Show  also   that  any  odd  number  whatever  may  be  represented 
by  2  n  -f- 1  by  giving  a  suitable  integer  value  to  n. 

4.  What  are  the  values  of  n^-\-  n  +  17  when  n  =  1,  2,  3,  •••,  9?     Are 
these  values  prime  or  composite  ? 

*  Compare  note  to  Ex.  15  below. 


54-65]       COMBINATORY  PROPERTIES   OF  NUMBERS  85 

5.  Do  the  answers  of  Ex.  4  warrant  the  conclusion  that  n^  +  n  +  17 
represents  a  prime  number  for  every  integer  value  of  n  (cf .  §  49)  ?  Is 
not  17  a  factor  oi  n^  +  n  + 17  when  w  =  17  ? 

6.  Do  the  expressions  x^  +  x  +  4:1  and  2  a:^  +  29  represent  prime  or 
composite  numbers  when  x  =  1,  2,  3,  •••  ?  Are  their  values  prime  for  all 
integral  values  of  x? 

Note.  The  above  questions  are  designed  to  emphasize  §  49  by  showing  the 
kind  of  errors  into  which  some  distinguished  matliematicians  have  been  led  by 
basing  general  conclusions  upon  more  or  less  numerous  verifications.  The  cele- 
brated mathematician  Fermat  concluded  from  a  certain  number  of  verifications 
that  2»  +  1  is  always  prime  when  n  =  2,  22,  2^,  2^,  ••• ;  Euler,  however,  discovered 
later  that  2^2  + 1  is  a  composite  number. 

7.  What  is  meant  by  saying  that  addition  is  a  commutative  opera- 
tion (cf .  §  50)  ?     That  it  is  an  associative  operation  ? 

Is  subtraction  commutative?  Multiplication?  Division?  Illustrate 
your  answer  in  each  case. 

8.  What  is  meant  by  saying  that  multiplication  is  distributive  with 
reference  to  addition  (cf.  §  39,  and  footnotes,  pp.  55  and  83)  ?  Can  you 
name  another  instance  in  which  one  operation  is  distributive  with 
reference  to  another? 

9.  Regarding  the  expression  — (a+J  — c+---)  as  —l'(a  +  h  —  c-\ — ), 
apply  the  distributive  law  of  multiplication  as  to  addition  to  prove  the 
correctness  of  the  principle  gi^en  in  §  33  for  removing  a  sign  of  aggrega- 
tion preceded  by  the  minus  sign. 

10.  By  means  of  the  commutative  and  associative  laws  of  multiplica- 
tion, show  that  (3  •  2)*  =  3*  .  2K     So,  too,  show  that  (a  •  6)"  =  a"  •  &«. 

Is  the  raising  of  the  product  of  several  factors  to  a  power  a  distributive 
operation  with  reference  to  the  factors  ? 

11.  Is  (2  +  5)2  equal  to  22  +  52?  Compare  this  with  Ex.  10,  and  then 
state  the  operations  over  which  an  exponent  is  distributive,  and  those 
over  which  it  is  not  distributive. 

12.  Which  of  the  combinatory  laws  discussed  in  the  present  chj 
is  it  usually  necessary  to  employ  when  a  polynomial  is  simplifit 
uniting  similar  terms?     When  a  polynomial  is  arranged  according" 
powers  of  one  of  its  letters  ?    When  an  equation  is  cleared  of  fractions  ?  * 

13.  Give  the  proofs  which  are  taught  in  arithmetic  of  the  principles 
given  in  §  54.  Compare  the  arithmetical  treatment  with  that  given  here, 
and  note  the  advantages  of  the  present  proofs. 

*  Compare  (1)  and  Ex.  10,  of  §  25. 


ing;  to 


86  ELEMENTARY  ALGEBRA  [Ch.  VI 

14.  Under  the  arithmetical  definition  is  —  a  fraction,  i.e.,  is  it  "  one 

n 

or  more  of  the  equal  parts  into  which  a  unit  has  been  divided  "  ?    How 

is  a  fraction  defined  in  the  preceding  pages  of  this  book  ?  Is  —  a  frac- 
tion under  this  definition  ?  ^ 

5 

15.  Write  down  the  successive  values  which  the  fraction  -  takes  when 

X 

the  values  1,  I,  |,  \,  -j^g,  •••  are  assigned  to  x.  How  do  these  successive 
values  of  the  fraction  compare  ?  Can  you  name  a  number  so  large  that 
none  of  these  values  of  the  fraction  will  exceed  it?  Can  you  name  a 
number  so  near  0  that  none  of  the  series  of  numbers  1,  |,  ^,  I,  y*g,  •••  will 
be  still  nearer  to  0  ? 

Note.  Ex.  15  illustrates  the  fact  that  in  mathematical  operations  numbers 
may  arise  which  are  greater,  and  others  which  are  less,  than  any  numbers  which 
we  can  name  or  even  think  of ;  such  numbers  are  usually  called  infinitely  large 
and  infinitely  small  numbers,  respectively,  —  all  other  numbers  being  classed 
together  as  finite  numbers.  An  infinitely  large  number  is  usually  represented  by 
the  symbol  oo. 

16.  Having  defined  0  as  a  —  a,  wherein  a  is  any  finite  number,  prove 
that  0  •  n  =  0  for  every  finite  value  of  n. 

Suggestion.  Substitute  a  — a  for  0,  then  apply  §  39,  and  finally  the  defini- 
tion of  zero. 

17.  Point  out  the  fallacy  in  the  following  reasoning : 
If  x  =  a, 

then  x'^  =  ax, 

and  x^  —  a^  =  ax  —  a% 

[Subtracting  a^  from  each  member 
i.e.,  '  (x  -{-  a)(x  —  a)  =  a(x  —  a) ; 

therefore  2  a(x  —  a)  =  a(x  —  a),  [Since  x  =  a 

«,  therefore,  2  =  1.  [Dividing  by  a  (a;  —  a) 


CHAPTER   VII 

TYPE   FORMS   IN  MULTIPLICATION— FACTORING 

I.     SOME  TYPE  FORMS  IN  MULTIPLICATION 

56.  Type  forms.  Although  all  exercises  in  multiplication  and 
division  of  integral  algebraic  expressions  can  be  readily  solved  by 
§  40  and  §  47,  yet  there  are  a  few  special  cases  of  these  operations 
which  occur  so  frequently  in  practice  that  it  is  well  worth  one's 
while  to  be  able  to  perforin  them  by  inspection ;  they  are  often 
spoken  of  as  type  forms.  Some  of  these  type  forms  are  considered 
in  the  next  few  paragraphs. 

57.  Square  of  a  binomiaL  This  may  be  divided  into  two  cases, 
according  as  the  binomial  is  the  sum  or  the  difference  of  two 
numbers. 

(i)  The  square  of  the  sum  of  two  numbers.  Let  a  and  6 
represent  any  two  algebraic  numbers ;  then  by  actual  multiplica- 
tion (§  40), 

-(a  +  b)(a-^b)  =  a'-j~2ab  +  b\  i.e.,  (a  +  by  =  a^-\-2ab-{-b'.* 

This  formula  may  be  translated  into  words  thus :  tlie  square 
of  the  sum  of  two  numbers  equals  the  square  of  the  first 
number,  plus  twice  the  product  of  the  two  numbers,  plus 
the  square  of  the  second  number. 

E.g.,  (x  +  3)2  =  a2-(-6a;-|_9  ;  Q/4.p)2  =  ^2  +  2yp+p2;  etc. 

(ii)   The  square  of  the  difference  of  two  numbers.     By 

actual  multiplication,  as  before, 

and,  in  general,  (a  —  by  =  a?  —  2  ab  +  6^.* 

The  student  may  translate  this  formula  into  words. 

*  This  second  member  is  called  the  expansion  of  the  binomial. 
87 


88  ELEMENTARY  ALGEBRA  [Ch.  VII 

Note.  If  either  or  both  of  the  terms  of  the  binomial  are  represented  by  more 
than  a  single  symbol,  they  may  be  inclosed  in  parentheses  (to  preserve  their 
individuality)  and  the  simplified  result  may  then  be  written  as  a  third  member 
of  the  equation. 

E.g.,     (2  X  +  3  2/)2  =  (2  x)2  +  2(2  x)  (3  y)  +  (3  ?/)2  =  4rX^  +  12zij +  9  y^. 

With  a  little  practice  and  care,  this  intermediate  step  may,  however,  be  safely 
omitted. 

EXERCISES 

Expand  the  following  expressions : 
1.   Qx  +  yy.  8.    (a -5)2.  ^^ 


a     xj 


2.    (w  +  n)2.  9.    (7-v)2. 

4.  (m  +  m>)2.  11.  (4a  +  7a;)2.  ^2a       3a;/ 

5.  (a  -  py.  12.  (3  m4  -  2  n)2.  17.  (9  aJc  +  hcdy. 

6.  (c-A)2.  13.  (|a:2-|)2.  18.  {{a  +  h)+cf. 

7.  (a;  +  3)2.  14.  (2  a^x  +  3  hy^y.  19.  {(a  +  &)  -  c}2. 

20.  Compare  the  fully  expanded  form  of  Ex.  18  with  (a  +  &  +  c)2, 
and  state,  if  you  can,  a  general  rule  for  writing  down  the  square  of  any 
trinomial  (see  also  §  61). 

21.  Expand  (x-y  +  z^- sy. 
Suggestion.    x--y  +  z  +  s={x  —  y)  +  {z  +  s). 

22.  Since  a  —  h=a-\-{  —  h),  show  that  case  (ii),  p.  87,  is  included  under 
case  (i). 

23.  Expand  (x^  +  y") 2.     Also  (3  a«  -  2  s'«)2. 

24.  What  must  be  added  to  a;2  +  6  a;  to  make  it  the  square  of  a;  +  3  ? 

25.  What  must  be  added  to  f^  +  H  to  make  it  the  square  of  <  +  |? 

26.  What  must  be  added  to  a*  +  0.%"^  +  ft*  to  make  it  the  square  of 
a2  +  62? 

27.  What  must  be  added  to  x^-\-2  x^y^  +  4  y^  to  make  it  the  square  of 
a;4  +  2  2/3  ? 

28.  Find  what  must  be  added  to  each  of  the  following  expressions  to 
make  them  exact  squares;  also  give  the  expressions  of  which  they  are 
then  the  squares : 

?n4-8m2n2  +  4n4;   a^-^ah;   x'^y^  +  12 xyz^ ',   x'^  +  ax\   ?^ndi  A^  +  -AB. 

n 

29.  Find,  by  the  method  of  §  57,  the  square  of  53,  i.e.,  of  50  +  3. 

30.  Write  down  the  squares  of  the  following  numbers :  18  (i.e.,  20—2), 
39,  71,  83,  and  34. 


57-58]  TYPE  FORMS  IN  MULTIPLICATION  89 

58.  Product  of  sum  and  difference.    If  a  and  b  represent  any  two 
numbers  whatever,  then,  by  actual  multiplication, 

(a  ^b)(a-b)  =  a'  -  b\ 

i.e.,  the  product  of  tTie  sum  of  any  two  numhers,  hy  the 
difference  of  these  numhers,*  is  tlxe  square  of  the  first  num- 
ber minus  the  square  of  the  second. 

E.g.,  (a;  +  3)(a;  — 3)  =  a'2-9;  (5  +  ?>i)  (5  —  m)  =  25  —  m2 ;  etc. 

Note.    Here,  too,  as  in  §  67,  complex  terms  may  be  iticlosed  in  parentheses, 
thus : 

(3x2  +  5^/)  (3x2-5?/)  =  (3 a:2)2 -  (5 y)2  =  9x4-25^2. 

/ 

EXERCISES 

Without  actually  performing  the  following  indicated  multiplications, 
write  down  the  products  by  inspection  : 

1.  ix^y-){x-y).  8.  (x^  +  ?/2) (^3  _  ^^2) . 

2.  (m  +  n)(m-n).  9.  {\^lmn-^'p\^){\^lmn^-^'p\^). 

3.  (3  a; +  2/)  (3  a;-?/).  10.  {^x  -  y^  ^  z}{{x  -  y)  -  z\. 

4.  (J  x-2y)Qx^1y).  11.  {{a? J^y^) - ab^iia^ ^V') ^ ah}. 

5.  (14a+15&)(14a-15&).  12.  («  +  6  +  c)(a  +  &- c). 

6.  (6jo-5^)(6/>+59).  13.  {a-h^c^ia-h-c). 

7.  (4m2-3n3)(4m2+3n3).  14.  (a  -  &+ a:)(a  +  J  -  x). 

15.  (m  -  2  n  +  s  -  0  (w  -  <  +  2  n  -  s). 

16.  Show  that  a:2  +  2  x?/  +  ?/^  —  2^  is  the  product  of  the  sum  and  differ- 
ence of  X  +  2/  and  z. 

17.  Show  that  a^  -{■  2  ah  -^  V^  -  c^  —  1  cd  -  d^  \s,  the  product  of  the  sum 
and  difference  of  a  +  6  and  c-\-  d. 

18.  (9a;2-42/2)-(3a;-2?/)  =  ?     Why? 

19.  (16a2-25&2)-^(4a  +  5J)=?     Why? 

20.  (a;4 _  ^/4)  ^  (a;2 _  ^2)  ^  ?     Why? 

21.  (a:6  _  1/4)  ^  (^8  _  2/2^  ^  ?  22.    (xis  -  3/8)  ^  (a:9  +  i/4)  =  ? 
23.   Find,  by  the  above  method,  the  product  of  22  by  18. 
Suggestion.    22  =  20  +  2  and  18  =  20  — 2. 

*  The  order  in  which  these  numbers  are  written  being  the  same  in  both  factors. 


90  ELEMENTARY  ALGEBRA  [Ch.  VII 

24.  By  this  method  find  the  following  products :  63  by  57 ;  48  by  52 ; 
34  by  26. 

Note.  The  identity  (a  +  6)  (a  -  6)  =  cfi -  62,  i.e.,  a2  =  (a  +  &)  («-&)  +  6^, 
furnishes  a  very  practical  device  for  mentally  squaring  any  number  consisting  of 
two  digits. 

E.g.,  to  square  73  mentally,  let  a  =  73  and  6  =  3;  then  the  last  formula  above 

^^*^°°^®^  (73)2  =  76  .  70  +  9  =  5329. 

Similarly,  to  square  58,  let  a  =  58  and  6  =  2;  then  the  formula  becomes 
(58)2  =  60.56  +  4  =  3364. 

25.  By  the  method  given  in  the  above  note,  write  down  the  square  of 
47 ;  of  82  ;  of  29 ;  of  53 ;  of  98 ;  and  of  61. 

59.  Product  of  binomials  having  common  term.  By  actual  mul- 
tiplication, 

(a;  +  3)(a;  +  5)  =  a;2_^8a;  +  15  =  a;2_|_(3^5)^_|.15. 

and        (a;  +  3)(a;-5)  =  a^-2a;-15  =  ar^+(3-5>-15. 

So,  too,  in  general,  (x  -{•  a)  (x -\- h)  =  x^ -\- {a -{-  h)x  -|-  ah ; 

i.e.  the  product  of  two  binomials  having  a  term  in  common 
equals  the  square  of  the  common  term,  plus  the  algebraic 
sum  of  the  unlike  terms  multiplied  by  the  common  term, 
plus  the  product  of  the  unlike  terms. 

EXERCISES 

Without  actually  performing  the  following  multiplications,  write  down 
the  products  by  inspection : 

1.  (a+5)(a  +  7).  10.  {a  +  h){a  +  c). 

2.  (a-5)(a-7).  11.  {a-h){a  +  c). 

3.  (a+5)(a-7).  12.  (2  a;  +  3)(2  x- 5). 

4.  (a-5)(a  +  7).  13.  (3  a  +  4)(3  a- 6). 

5.  (y_c)(2/  +  2c).  14.  (4rt2_5)(4a2+ 1). 

6.  (a:2  +  4)(a:2+5).  15.  (xy- 4)(a:7/+ 16). 

7.  (a:2+4)(a:2-5).  16.  Q^mH^ +  2)(lhnhi^ -d>). 

8.  (a:2-4)(a;2_5).  17.  {Q  ^  m)  -  2}{{l  +  m)  -  b}. 

9.  (a;2-4)(a;2+5).  18.  {(/  +  ,«)+ 8}{(^  +  m)  -  15}. 


58-61]  TYPE  FORMS  IN  MULTIPLICATION  91 

60.   Product  of  two  binomials  which  contain  the  same  letters. 

The  product  of  two  binomials  containing  the  same  letters  is  a 
trinomial  which,  by  a  little  practice,  may  be  written  down  without 
writing  the  intermediate  steps. 

E.g.,  the  product  of  3  a;  +  5  and  2  cc  —  7  may  be  arranged  as  in  the  margin :  the 
term  (5  x'^  is  the  product  of  the  first  terms  of  the  binomials,  the  term  — 11  a;  is  the 
algebraic  sum  of  the  "  cross  products  "  (2  cc  by  5  and  3  a;  by  —  7) , 
and  —35  is  the  product  of  the  last  terms  of  the  binomials.        3x  +5 
This  final  product  may,  with  a  little  practice,  be  easily  written        2  a;  —  7 
down,  omitting  the  intermediate  steps.  i-UAn 

Similarly,  in  the  product  of3x  +  4?/  by  5a  —  2 y,  the  prod-  _         _ 

uct  of  the  first  terms  is  15  x^,  the  algebraic  sum  of  the  cross        

products  is  14  xy,  and  the  product  of  the  last  terms  is  —  8  2/2 ;        6  a;2  — 11  a:  —  35 
hence  (3  a;  +  4  ?/)  (5  a;  —  2  ?/)  =  15  x^  + 14  x?/  —  8  y^.      So,    too, 
(ax  +  6)  (ex  +  (^)  =  acx2  +  (ad  +  6c)x  +  6(^. 


EXERCISES 

Write  down  the  following  products  by  inspection : 

1.  (3a;+2)(4a;-3).  5.    (7  0^+ 62)(3  ^2+ 8  &2). 

2.  {^x  +  2y){4.x  +  Zy).  6.    (Q  x -2  y){x+ y). 

3.  (a:-3y)(5a:+6y).  7.    (x  +  d){x+h). 

4.  (2a-4&2)(5a-6  62). 

61.   The  square  of  any  polynomial.     By  actual  multiplication  it 
is  found  that 

(ct  +  6  +  c)2  =  a2  +  62_,_c2_p2a6  +  2ac  +  2&c, 
{a  +  h ■\-  G-\- df  =  a?  +  h^  +  (? -\- d}  +  2  ab ^2  ac  +  2 ad  +  2hc 
+  2hd  +  2cd, 
(a.+ 6  4- c  +  d+e)2  =  a2  _,_  52  _|_c2  ^^2  ^  g2^  2  a6  + 2  ac  + 2  ac^  + 2  ae 
+  2  6c  +  2  &d  4-  2  &e  +  2  cd  +  2  ce  +  2  de, 

etc.  This  may  be  formulated  into  words,  thus :  the  square  of 
any  polynomial  whatever  equals  the  sum  of  the  squares 
of  all  the  terms  of  the  polynoinial,  plus  twice  the  product 
of  each  term  hy  all  the  term^s  that  follow  it* 

*  The  formal  proof  of  this  theorem  is  given  in  Chapter  XVIII. 


92  ELEMENTARY  ALGEBRA  [Ch.  Vll 

EXERCISES 

Expand  the  following  expressions  by  inspection : 

1.  (m+n-.s)2.  a    (rt_6  +  c-rf)2 

2.  (a-&-c)2.  9.    (ax  +  hy  +  czy. 

3.  (2x  +  y  +  2)2.  10.    {ahx  -  acy  -  hczy\ 

4.  (2  a;  +  3  ?/  -  2;)2.  11.    (^  +  /^j  +  n  +  p  +  (^  +  r  +  s)2. 

5.  {2x-^y  +  zy.  ^         12.    (2x-3  2/  +  4  2-5a+3  6-4)2. 

6.  (3  a  +  4  &  +  c)2.  13.    (a;4  +  2  a:8  -  3  a:2  +  4  a;  -  5)2. 

7.  (3a-46-2c)2. 

62.  Higher  powers  of  binomials  —  binomial  theorem.  By  actual 
multiplication  it  is  found  that 

(x  -\-  yy  =  x^  -\-  4.:»?y  +  Q ^y"^  ^^xf-\-  y\ 

(a;  +  ?/)«  =  a^  +  5  x'^y  +  10  a^?/^  +  10  x^  +  6xy^-\-  f, 

{x  +  yf  =  x^  +  Qx^y  +  l^xy  -\-20  ^f  +  15xY  +  Qxf  -^y"^, 

etc. ;  and  that 

{x  -  7jy>=  a?  -  ^x'y  +  ^xy'-  f, 

{x  —  yy  =  x^  —  4:a:^y-\-6  a?y'^  —  4  a^?/^  +  y^, 

{x  —  7jy  =  x^~5  x^y  +  10  a^2/^  —  10  x^y^  -{-5xy*  —  f',  etc. 

A  careful  study  of  the  second  members  of  the  above  equations 
will  show  that  they  all  follow  the  same  laws,  and  that  they  may, 
therefore,  be  written  down  by  the  same  rules.  In  fact,  such  a 
study  will  show  that : 

(1)  The  number  of  terms  in  the  expansion  is  in  every 
case  greater  by  1  than  the  exponent  of  the  binomial. 

(2)  The  X  *  appears  in  every  term  of  the  expansion  except 
the  last,  and  the  y  appears  in  every  term  ^  the  expan- 
sion except  the  first. 

(3)  The  exponent  of  x  in  the  first  term  of  the  expansion 
is  the  same  as  the  exponent  of  the  binomial,  and  it  decreases 
by  1  from  term  to  term  in  passing  to  the  right,  while  the 

*  In  applying  these  rules  to  other  hinomials,  observe  that  cc  is  here  used  for 
"  the  first  term  of  the  binomial "  and  y  for  "  the  second  term  of  the  binomial." 


\ 


61-62]  TYPE  FORMS  IN  MULTIPLICATION  93 

exponent  of  y  in  the  second  term  of  the  expansion  is  1,  and 
it  increases  hy  1  from  term  to  term  in  passing  toward  the 
right. 

(4)  The  coefficient  of  the  first  term  of  the  expansion  is  1 ; 
the  coefficient  of  the  second  term  is  the  same  as  the  exponent 
of  the  binomial ;  and  if  the  coefficient  of  any  term  he  mul- 
tiplied hy  the  exponent  of  x  in  that  term,  and  this  product 
he  divided  hy  the  /rtumher  of  the  term  (i.e.,  hy  this  term's 
exponent  of  y  increased  hy/T),  the  result  will  he  the  coeffi- 
cient of  the  next  term. 

(5)  The  signs  of  the  terms  of  the  expansion  are  all  posi- 
tive if  each  term  of  the  hinomial  is  positive,  hut  if  the 
second  term  of  the  hinomial  is  negative,  then  the  terms  of 
the  expansion  are  alternately  positive  and  negative —  the 
first  term  heing  positive. 

Note.  It  is  proved  later  (Chap.  XVIII)  that  the  above  laws  apply  to  all 
positive  integral  powers  of  any  binomial  whatever ;  hence  such  powers  may  be 
expanded  without  actually  performing  the  multiplications. 

Ex.  1.   Expand  (a  —  hy. 

Solution.  By  (1),  (2),  and  (3)  above,  the  letters  and  exponents  in 
the  several  terms  of  this  expansion  are : 

a^      a'b      a^b^      a%^      a^b*      a%^  •    a%^      ab'^      b^; 
by  (4),  the  coefficients  are : 

1         8        28        56        70        56        28         8         1; 
and  by  (5),  the  signs  are: 

+       -.+        -        +        -        +        -+; 
hence,  combining  these  results, 
(a  _  J)  8  zz:  a8  -  8  a^ft  +  28  a662  _  56  a^b^  +  70  a^b^  -  56  a%^  +  28  a^js  _  8  a&H  i^ 

Ex.  2.   Expand  (2  x  -  a^y. 

Solution.     Letters  and  exponents, 

(2  xy     (2  xy  (a2)     (2  x)  (a2)2     (a^)^ ;      [Cf .  (1),  (2),  (3) 
coefficients,  1  3  3  1 ;  [Cf.  (4) 

signs,  +  -  +  -  ;  [Cf.  (5) 

combined  result,  (2  x  -  a^y  =  (2  xy  -  3  (2  xy(a^)  +3(2x)  (a^y  -  (a^y ; 
simplified  result,  (2  x  -  a'^y=  8  x^  -  12  x^a^  +  6  xa*  _  a^. 

With  a  little  practice  the  combined  result  may  be  written  down  at 
once  instead  of  making  several  steps  of  the  work. 


94  ELEMENTARY  ALGEBRA  [Ch.  VII 

EXERCISES 

Expand  the  following  expressions : 

3.  (a +  6)3.  6.    (u-vy.  9.    (a:-y)io. 

4.  (a~xy.  7.    (a:  +  -)*-  10.    (x-2ay. 

5.  (m-ty.  8.    (3  a2- 2  65)3.  n.    (;n2  +  3n)6. 

12.  Write  the  first  4  terms  of  (a  +  x)^. 

13.  Write  the  first  3  terms,  and  also  the  7th  term,  of  (x  —  yY^. 

14.  Write  the  first  5  terms  of  (2  ax  -  3  k^y, 

II.    FACTORING 

63.  Definitions.  In  a  broad  sense,  any  two  or  more  numbers 
whose  product  is  a  given  number  are  factors  of  that  number. 

Thus,  since  -i-  •  |-  •  10  =  4,  therefore  \,  f ,  and  10  are  factors 
of  4;  so  also  are  j^,  18,  and  j%. 

In  this  sense,  however,  the  problem  of  finding  the  factors  of 
any  given  number,  or  algebraic  expression,  is  manifestly  inde- 
terminate; it  is  therefore  customary,  when  speaking  of  factors, 
to  mean  only  the  rational  *  and  integral  exact  divisors  of  a 
given  number  or  expression. 

E.g.,  ±l,t  ±2,  ±3,  ±4,  ±6,  and  ±12  are  factors  of  12;  and  ±1,  ±5, 
±  (2  a;  +  ?/) ,  ±  (2  a;  —  z/) ,  as  well  as  products  of  any  two  or  more  of  these,  are 
factors  of  20  x^  —  5  y^.  Every  number  is  a  factor  of  itself,  and  1  is  a  factor  of 
every  number. 

A  number,  or  an  algebraic  expression,  is  said  to  be  prime  if  it 
has  no  exact  divisor  {i.e.,  factor)  except  itself  and  unity;  other- 
wise it  is  composite. 

A  factor  is  prime  or  composite  according  as  the  expression  for  it 
is  prime  or  composite ;  and  it  is  integral  with  regard  to  any  given 


*  An  expression  is  rational  with  regard  to  a  particular  letter  if  it  cpntains  no 
indicated  root  of  that  letter  (see  §  130) . 

t  The  sign  db  is  called  the  double  sign,  and  is  read  "  plus  or  minus  "  ;  it  is  used 
to  combine  two  expressions  into  one :  thus  the  expression  ±  a  means  both  +  «  and 
also  —  a. 


62-66]  FACTORING  95 

letter  if  the  algebraic  expression  for  it  is  integral  with  regard  to 
that  letter  (cf.  §  41). 

It  will  appear  later  that  the  writing  of  an  expression  as  the 
product  of  its  prime  factors  often  greatly  simplifies  algebraic 
work ;  and  it  is  therefore  important  that  the  student  should  early 
master  those  cases  of  factoring  which  present  themselves  most 
frequently.     Some  of  these  cases  will  now  be  considered. 

64.  Factors  of  a  monomial.  This  is  the  simplest  of  all  the  exer- 
cises in  factoring,  and  can  be  done  by  inspection. 

E.g.,  30  ax'^y  =  2  -^ -^  •  a-  x  -x  -y,  which  exhibits  the  given  monomial  as  the 
product  of  its  prime  factors;  the  product  of  any  two  or  more  of  these  prime 
factors  is  a  composite  factor  of  the  given  monomial  (cf .  §  63)\ 

A  rule  for  this  kind  of  factoring  may  be  stated  thus :  by  inspec- 
tion, or  hy  trial,  find  the  prime  factors  of  the  numerical 
coefficient  of  the  given  rnonomial,  and  to  their  indicated 
product  annex,  each  of  the  literal  factors  as  many  times 
as  there  are  units  in  its  exponent, 

EXERCISES 

Separate  the  following  monomials  into  their  prime  factors : 

1.   Qa^x^  2.    15  mY^^-  3.   36  sHK 

4.   420  m%V-  5.    572  a^c^uv^. 

65.  Monomial  and  polynomial  factors  of  a  polynomial.  If  a  poly- 
nomial contains  a  monomial  factor,  the  latter  can  usually  be  dis- 
covered by  mere  inspection. 

E.g.,  in  12  a^^  +  4  abx'^  —  8  axhj'^,  it  is  seen  that  each  term  contains  the  factor 

^        '  12  a'hfi  +  4 ahxhj  —  8  axhj^  =  4  oa;2  .  {^ax-\-by  —  2y^. 

In  order  to  factor  a  polynomial  completely,  it  is  then  only 
necessary  to  consider  further  how  to  factor  a  polynomial  which 
contains  no  monomial  factor.  This  problem,  however,  is  in  general 
very  difficult,  and  only  its  simplest  cases  will  at  present  be  con- 
sidered. Fortunately  it  is  these  simpler  cases  which  present 
themselves  most  frequently  in  practice. 


1. 

5a-106. 

2. 

17  x^  -  289  xK 

3. 

4  a:3  -  8  x'^y. 

4. 

10  m%2  _  15  ,w8„8. 

5. 

lQx'^-2ahx. 

6. 

4  a%'^  -  24  a258. 

7. 

15  a:*  _  10  a;3  +  5  a;2. 

8. 

3  a^  -  6  a-ife  +  a4^,2. 

9. 

:ci2^i2  +  a;iYi  + 3,103,8  _ 

.0. 

3  7715  _  12  mH'^  +  6  mn*. 

96  ELEMENTARY  ALGEBRA  [Ch.  VII 

EXERCISES 

Separate  the  following  expressions  into  their  monomial  and  poly- 
nomial factors : 

11.  ac  —  be  —  cd  —  abed. 

12.  13  xY  -  13  xY  +  12  xy. 

13.  14  xYz^  -  7  xYz^  +  8  xy^z^. 

14.  60m2nV2-45m%V  +  90m%V2. 

15.  12  x'^b^y  -  18  xy%  +  24x4&4^4. 

16.  14  ahnn^- 21  a^m^^-^9  a^mn^. 

17.  25  c'^dx^  +  35  03^2^:4  _  55  c2</2^5. 

18.  51  0:3/2^3  _  68  xY^^  +  85  x^y^z*. 

19.  52  a2^,8c4  _  65  a^^c"^  +  91  a2J2c2. 

20.  44  a^xY^  +  66  a^xY  +  88  a2a;5j,4. 

66.  Use  of  type  forms  in  factoring.  Since  finding  the  factors  of 
a  given  number  or  expression  is,  in  a  certain  sense,  the  undoing 
of  a  multiplication,  therefore  the  type  forms  in  multiplication 
already  studied  (§§  57-62)  may  be  advantageously  employed  in 
separating  certain  types  of  expressions  into  their  factors ;  some  of 
these  will  now  be  given. 

(i)  Trinomials  of  the  type  x^  ±2xy  -\-  y^*  In  §  57  (i)  and 
(ii)  it  is  shown  that,  whatever  the  numbers  or  expressions  repre- 
sented by  a  and  b, 

(a  +  by  =  a'-\-2ab  +  b^  and   (a-bf  =  a" -2ab +  b^; 

therefore  a-\-b  and  a  +  &  are  the  factors  of  a^  +  2ab  -\-  b^,  and 
a  —  ft  and  a  —  6  are  the  factors  of  a^  —  2  a6  +  b^. 

Similarly  in  general,  if  in  a  trinomial  two  terms  are  exa/it 
squares,  and  the  remaining  term  is  the  double  product  of 
their  square  roots  A  then  the  given  trinomial  is  the  square 
of  a  hinomidt. 

E.g.,  m2 -f  6  mn  +  9  n"^  is  a  trinomial  of  this  type,  and  its  factors  are  ?n  +  3 n 
and  ??i  +  3  n ;  so,  too,  is  4  a;2 -}_  25  —  20  x,  of  which  the  factors  are  2  a;  —  5  and  2  x  —  5. 

*  x2  ±  2  x?/  +  ?/2  means  both  x^  +  2  x?/  +  7/2  and  also  x2  —  2  x?/  +  7/2  (cf .  §  63, 
footnote). 

t  The  square  root  of  a  number  is  that  number  which,  being  multiplied  by  itself, 
will  produce  the  given  number.    Cf .  §  122. 


\. 


66-66]  FACTORING  97 

EXERCISES 

Factor  the  following  expressions : 

1.  a:2-6x  +  9.  3.    1  -  \0 y  i- 25 y"^.  5.   x^-4:x^  +  4:. 

2.  225  +  30  a;  +  x^.  4.   x-  +  4:xy-\-4:  y'^.  6.   a^^  +  2  at  +  1 . 

7.  What  first  suggests  to  you  that  x-  +  9y^  -\-  6  xy  may  be  the  square 
of  a  binomial?  How  do  you  test  the  correctness  of  this  supposition? 
When  is  a  trinomial  the  square  of  a  binomial? 

8.  Write  out  a  carefully  worded  rule  for  factoring  expressions  of  the 
type  x^±2xy  +  y^.  How  are  the  terms  of  the  binomial  obtained  ?  How 
determine  the  sign  by  which  they  are  to  be  connected  ? 

9.  Is  a*  +  2  a%^  —  b^  the  square  of  a  binomial  ?     Why? 

10.  Is  (x  +  y)2  +  («  +  6)2  4-  2(a  +  b)  (x  +  y)  the  square  of  a  binomial  ? 

Separate  the  following  expressions  into  their  prune  factors,  and  check 
your  work  by  assigning  simple  numerical  values  to  the  letters  involved 
(cf.  Ex.  7,  §  39)  : 

11.  a^b^  +  6  abed  +  9  c^d^.  13.    9  x^  -  12  xyz  +  4:  yh^. 

12.  4x4-64x2  +  256.  14.   81  x2  -  18  ax  +  a^. 

15.  196  a2&2c2  +  112  ab^c^d  +  16  b^c^d^. 

16.  yi-^y(x  +  y)+4:(x  +  yy. 

17.  (X  +  yy  -  10(x  +  2/) (y  +  2)  +  2o(y  +  zy. 

18.  16(a  +  x)2  -  32(a  +  x) (x-y)-\-  16(x  -  y)^. 

19.  25(x  +  yy-50(x  +  y)z*-\-25z^ 

20.  4(a  +  3  &)2  -  24(a  +  3  &)(6  -  c)  +  36(6  -  c)2. 

21.  9  a^**  -  12  a~62»  +  4  64».  23.    -  x*  +  2  a2x8  -  a%2. 

22.  -x^-lQy^-8xY-  24.    (x2  +  ?/2)2_  2(a;2+ ?/2)22  +  ^4. 

(ii)   Expressions  of  the  type  x^  —  y^.     In  §  58  it  was  shown 
that,  whatever  the  numbers  or  expressions  represented  by  a  and  h, 
(a+6)(a-&)  =  a2-62; 

therefore  the  factors  of  a?  —  h^  are  a-\-h  and  a  —  6. 

Similarly  the  difference  of  the  squares  of  any  two  numbers  or 
expressions  may  be  factored. 

B.g.i  25  n2  —  9  <2  is  of  this  type,  and  its  factors  are  5  n  +  3  «  and  5  n  —  3  f ;  so, 
too,  a2  +  62  and  a2  —  62  are  factors  of  a*  ~  6*,  but  a2  —  62  is  not  prime ;  the  prime 
factors  of  a*  —  6*  are  a2  +  62,  a+b,  and  a  —  b. 


6. 

25x^-9f. 

11.    121a4-36&i 

7. 

ai6-4&8. 

12.   64xy«-144z2. 

8. 

a^x  -  hH. 

13.    (a:  +  2/)2_(a  +  c)2. 

9. 

36aV_81rf2. 

14.   49-36a:V. 

.0. 

a:2«  -  4. 

15.     w2„_^2m. 

\ 

18.   289a;c^9-/"s. 

19.    16</2_ 

-9(x-2/)2. 

98  ELEMENTARY  ALGEBRA  [Ch.  VII 

EXERCISES 
Factor  the  following  expressions : 

1.  y^-z^. 

2.  2/2  _  9^2. 

3.  4^2_25&2. 

4.  225a262-16. 

5.  9y2_l. 
16.,  169  xYz^  -  IQ  y^d^ 
17.   324a:2?/426_81. 

20.  For  what  values  of  a  and  &  is  (a  +  ft) (a  —  b)  equal  to  a^  —  h^l  Is 
this  equation  true  even  when  a  =  b?     (Cf.  §  55.) 

21.  Factor  a^  +  2ab-  c'^  +  b\ 

Suggestion.    a2  +  2  a6  —  c2  +  62  =  a2  +  2  a&  +  &2  _  c2  =  (a  +  &)2  —  c2. 

By  rearranging  and  grouping  the  terms  as  in  Ex.  21,  factor  the 
following : 

22.  62^2&c  +  c2- J2.  28.  4a2+ 962_  ig  ^2  _  12a&. 

23.  a2-6aa:+9a;2-4c2.  29.  9 a:2  -  25 32  +  16 ^^2  +  24 a:y. 

24.  a^-\-2ab-d'^-\-b\  30.  ^.b"^ -  x'^^  ^xy -^  o?-^-^ah-^y\ 

25.  (a;  +  2/  +  e)2_a2-2a6-&2,      31.  l-x'^-1xy-y\ 

26.  x2  -  J2 _  2 ^2/ +  2/2.  32.  1 -4a: +  4x2-1  + 6a: -9x2. 

27.  x2  + 4x^-4 22 +  4^2.  33.  2562-1 -9 &2a;2_i0aj+a2+6&x. 

(iii)  Expressions  of  the  type  x^  +  {a-[-h)x-{-  ah.  In  §  59  it 
was  shown  that,  whatever  the  numbers  or  expressions  represented 

by  a,  b,  and  x, 

(x  +  a)  (x  -{-  b)  =  a^  -\-  (a  -{-  b)x  +  ab. 

This  formula  is  helpful  in  factoring  trinomials  of  the  above  type. 

E.g.,  x2  +  7  K  + 12 *  may  be  written  cc2  +  (4  +  3)  x  +  4  •  3 ;  it  is  therefore  of  this 
type,  and  its  factors  are  x  +  4  and  x  +  3. 

Observe  that  the  plan  of  factoring  this  trinomial  is  first  to  find  all  the  pairs  of 
numbers  whose  product  is  12,  then  to  select  from  among  these  that  pair  whose 
sum  is  7 ;  from  which  the  required  factors  are  manifest. 

In  the  same  way  it  may  be  shown  that  the  factors  of  m2  —  6  m  +  8  are  m  —  4 
and  m  — 2;  so,  too,  x2  + 2x -  15  =  (x +  5) (x  — 3) ;  9y2  —  lSy  —  7,  i.e.,  (3y)2 
—  6(3?/)— 7=  (3?/  — 7)(3y  +  l)  ;  and  x2- 3nx  — 28  a2  =  (x +  4a)  (x  — 7a). 

This  method  of  factoring  expressions  of  the  form  x2  +  ax  +  6  is,  however, 
advantageous  only  when  the  number  of  pairs  of  factors  of  b  is  not  large ;  another 
method  is  given  in  §  164,  Ex.  69. 

*  Such  an  expression  is  usually  called  a  quadratic  trinomial. 


66]  FACTORING  99 

EXERCISES 

1.  If  the  expression  x^  +  6  x  —  SQ  is  the  product  of  two  binomial 
factors,  what  is  the  product  of  the  unlike  terms  in  these  binomials  ? 
Have  these  terms  like  or  unlike  signs  ?  Why  ?  What  is  the  sum  of  these 
unlike  terms  ?    Is  the  larger  of  them  positive  or  negative  ?    Why  ? 

2.  Based  upon  such  considerations  as  those  given  in  Ex.  1,  write  out 
a  carefully  worded  rule  for  factoring  trinomials  of  this  type. 

Separate  each  of  the  following  expressions  into  its  prime  factors  : 

3.  a;2  -  3  a;  +  2.  7.    a2  -f  7  a  -  30.  11.    Q  y  -  y"^  -  y\ 

4.  x'^  +  x-Q.  8.    n2-4n-60.  12.   a:^  -  17  a;2  +  72  a:. 

5.  x'^-x-2.  9.  p2  _  12  p  +  35.         13.    13  a;  -  30  +  x\ 
6.2/2-6^  +  5.               10.   4  -  6  a:  +  2  a;2.  14.   x^  -  24  a;^  +  63. 

15.  3  3/6  +  39  ?/3  +  66.  18.   a:-2  -  26  a:-i  +  69. 

16.  a;2  +  (3  a  -  2  &)  a;  -  6  ah.  19.   a^"^  -  7  ab -{-  10. 

17.  ax2  +  7  a2a;  +  6  a^.  20.    (a:  +  2/)2  +  7  (a:  +  ?/)  +  6. 

21.  9a;2+6a:-8.     Suggestion.    9a;2  +  6x  — 8=  (3x)2  +  2(3x)  — 8. 

22.  4  ar2  +  4  a;y  -  3  y^  24.    15  x^  +  32  x^y  +  16  xY- 

23.  16  a;2  +  32  a;  +  15.  25.   a;^^  +  5  a:'*  +  6. 

26.  Can  a;2  +  a:  +  6  be  separated  into  two  binomial  factors  like  those 
found  for  the  other  exercises  above  ?     Explain. 

(iv)  Expressions  of  the  type  acoi?  +  (ad  ■\-hG)x-\-  bd.  The 
foregoing  method  is  easily  extended  so  as  to  include  many  tri- 
nomials which,  are  not  of  type  (iii). 

From  §  60  it  follows  that  if  the  trinomial  6  a?^  —  11  ic  —  35,  for 
example,  is  the  product  of  two  binomial  factors,  then  the  first 
terms  of  these  binomials  are  factors  of  6  a^,  and  the  last  terms  are 
factors  of  —  35 ;  hence  the  possible  pairs  of  binomial  factors  are : 

3a;-71 


6x-5]     (6x+5\     (6x-7}     (6x+7\    J3a;-5 
x+7j'\    a^-Tj'  I     a;+5r  t    x-5J'  \2x+7 


[2x+oj' 


etc. ;  and  from  among  these  the  pair  to  be  selected  is  that  one  for 
which  the  algebraic  sum  of  the  "  cross  products  "  is  —  11  a; ;  this 
pair  is  3aj  +  5  and  2  a;— 7,  hence  6  a;^— 11  a;— 35  =  (3a;-f5)(2  a;— 7). 
Similarly  it  is  found  that  12  a;^  +  8  a;  -  15  =  (6  a;  -  5)  (2  a;  +  3), 
and  that  15  a-  +  14  a&  -  8  6^  =  (3  a  +  4  6)  (5  a  -  2  b). 


100  ELEMENTARY  ALGEBRA  [Ch.  VII 

EXERCISES 

Factor  each  of  the  following  expressions : 

1.  Sx^  +  X-  10.  5.    16  a;5  +  4  xY  -  30  xy^. 

2.  4  a;2  +  16  a;  +  15.  6.   4  ab^  -  73  ahc  +  18  ac^. 

3.  Sy^  -lOxy  -d  x"^.  7.   90  xyz^  -  98  a^xyz  +  8  a^xy. 

4.  8  ^2  +  23  ^B  -  3  ^2.  8.   15  M*'  +  16  M2*iV2  +  4  iV*. 

•       9.   3  (a  +  6)2  +  10  (a  +  &)  (a  +  2  6)  -  8  (a  +  2  6)2. 

(v)  O^/z^r  types;  exact  powers.  The  formulas  of  §§  61  and 
62  may  also  be  employed  to  factor  polynomials  of  the  types  to 
which  they  belong.  When  such  polynomials  present  themselves 
for  factoring,  which  is  comparatively  seldom,  the  student  need 
only  arrange  them  properly  and  observe  whether  all  the  require- 
ments stated  in  §  61  or  §  62  are  satisfied ;  if  so,  the  given  poly- 
nomial is  an  exact  power,  and  its  factors  are  written  by  inspection. 

E.g.,  to  factor  the  expression  x^  +  z^  — 4:  yz-^  2  xz  +  iy^  — 4:  zy,  obserye  that 
it  consists  of  three  square  terms,  and  of  three  double  products,  hence  it  may 
belong  to  the  type  considered  in  §  61.  A  slight  rearrangement  of  the  terms  shows 
that  it  is  of  this  type,  viz.,  x^  +  'iy^^  z^  —  4:Xy +  2zz  — 4:yz=  (x  —  2y -{■  z)^. 
Similarly  for  expressions  which  belong  to  the  type  considered  in  §  62,  namely, 
powers  of  binomials. 

EXERCISES 

Factor  the  following  expressions,  and  check  your  results : 

1.  m2  -  2  ms  -  2  ns  +  s2  +  2  mn  -1-  n2. 

2.  y^  +  4:xy-{-4:X^  +  4:Xz-{-2yz  +  zK 

3.  m^-i^+dmt^-^m'^t. 

4.  a:4+8a;2+24  +  ^  +  ^. 

a;2      x^ 

5.  9  a2  +  4  c2  -  12  ac  +  16  &c  -  24  ab  -\- 16  b^. 
^^        6.    9m4+30m8  +  25m2-12TO2n4-4n2-20mn. 

67.  Factoring  by  means  of  the  remainder  theorem.    In  §  48  it  was 

proved  that  if  Ax''  +  Bx""-^  +  •••  +  Bx H-  ^  is  exactly  divisible  by 
x  —  a,  then  Aa""  +  Ba"-'^ -\-  •••  -\-Ha  +  K=0,  and  conversely;  on 
this  fact  is  based  a  simple  method  for  finding  binomial  factors  of  a 
large  number  of  algebraic  expressions. 


66-67]  FACTORING  101 

E.g.,  to  ascertain  whether  x  —  2  is  a  factor  of  x^  —  5z  +  6,  it  is  only  necessary 
to  substitute  2  for  a;  in  a;2  _  5  ^  +  6,  and  observe  whether  or  not  the  result  is  0 ; 
this  result  is  0,  and  therefore  a:  —  2  is  a  factor  oix^  —  5x  +  6. 

So,  too,  X  —  6  is  a  factor  of  a;2  _  g  a;  + 12  because  6^  —  8  •  6  + 12  =  0 ;  and  x  + 1, 
i.e.,  X  -  (- 1) ,  is  a  factor  of  x2  +  7  x  +  6  because  (-  1)2  +  7(-  1)  +  6  =  0. 

Again,  if  a:  —  a  is  a  divisor  of  x^  —  2  x2—  9  x  + 18,  then  18  is  the  product  of  a  by 
the  last  quotient  term ;  hence,  in  seeking  this  class  of  factors  of  x^  —  2  x2  —  9  x  + 18, 
only  numbers  which  are  factors  of  18  need  be  tried  in  the  place  of  a.  The  factors 
of  18  are:  +1,  —1,  +2,-2,  +3,-3,  +6,  —6, +9,  —9,  +18,  and— 18;  if  these 
numbers  be  substituted  in  turn  for  x  in  the  given  expression,  it  will  be  found  that 
+  2  is  the  first  one  that  reduces  that  expression  to  0,  therefore  neither  x  —  1  nor 
X  + 1  are  factors,  but  x  —  2  is  a  factor ;  further  trial  will  show  that  x  —  3  and 
X  +  3  are  also  factors  of  the  given  expression. 

When  any  factor  of  an  expression  has  been  discovered,  by  any 
process  whatever,  that  factor  may  be  divided  out  of  the  given 
expression,  and  the  remaining  factors  may  then  be  more  easily 
found. 

EXERCISES 

1.  If  X*  +  Q  x^  —  I2x  +  5  be  divided  by  x  —  a,  what  will  be  the 
remainder?  Without  performing  the  division,  find  the  remainder  when 
the  divisor  is  a:  —  2  (of.  §  48);  also  when  it  is  a;  +  1,  and  when  it  is  x  —  1. 
Which  of  these  divisors  is  a  factor  of  the  given  expression  ? 

2.  If  the  expression  x^—dx^  —  x-\-d  has  a  factor  of  the  form  x  —  a, 
what  are  the  four  possible  values  of  a  ?  Find  all  such  binomial  factors  of 
a;3  -  3  a;2  -  a:  +  3. 

By  the  above  method,  find  all  the  factors  you  can  of  the  following 
expressions : 


3. 

xs-'jx  +  e. 

8. 

mj4-15m;2  +  10i^  +  24. 

4. 

a:^  -  9  a;2  +  23  x  -  15. 

9. 

a8  +  7  a2  +  2  a  -  40. 

5. 

a:3  +  14  a;2  +  35  a:  +  22. 

10. 

c8  _  5  c2  -  29  c  +  105. 

6. 

x3-lla:2  +  31a;-21. 

11. 

x^-x^-7  x^+x+Q. 

7. 

p  +  4  p  _  11  ;^  _  30. 

12. 

y5_iOj,4  +  40  3/3_80z/2  +  80 

32. 

13.  If  a;  —  A:  is  a  factor  of  any  given  expression,  what  does  the  value 
of  that  expression  become  when  x  =  k'i  Why?  Prove  that  the  converse 
of  this  is  also  true. 

14.  By  means  of  the  remainder  theorem  show  that  a  —  h,!)  —  c,  and 
c  -  a  are  factors  of  a(&2  _  c^)  +  &(c2  -  a^)  +  c{a^  -  b^). 


102  ELEMENTARY  ALGEBBA  [Ch.  VII 

15.  What  is  the  remainder  when  (2  a:  —  3  a) 2  +  (3  a?  —  a) ^  is  divided 
by  a:  —  a  ?    When  (z  —  y  +  2)^  —  y^  -\-  x^  is  divided  by  x  —  y'i  by  x+yl 

16.  Find  the  factors  of  4  x^  -  4  x^  -  9  a;  +  9. 

Suggestion-.  ^x^  —  ^x^  —  Qx-\-^  =  ^{z^—z^—^iX-\-^)\  now  apply  the  above 
method  to  the  expression  within  the  parenthesis. 

Find  the  factors  of : 

17.  4  a;2  -  16  a;  +  15.  18.    2  ?/8+ 5  ^2  _  2  ^  _  5. 

19.  What  value  of  x  will  reduce  to  zero  any  expression  which  contains 
2  X  —  d  as  a  factor?  How  then  may  the  remainder  theorem  be  used  to 
detect  the  factor  2  a;  —  3  in  any  given  expression  ?  Use  this  suggestion 
to  solve  Exs.  17  and  18. 

20.  What  is  the  remainder  when  x^  —  a"  is  divided  by  a:  —  a  ?  Why  ? 
When  a;"  —  a"  is  divided  by  a;  +  a  and  n  is  an  even  positive  integer? 

21.  Prove  that  a:  —  1  is  a  factor  of  every  expression  of  the  form 
^a;"  +  jBa;"-i  +  Cx"^-^  +  •••  +  Hx  +  iC  =  0,  in  which  the  sum  of  the  positive 
coefficients  (among  A,  B,  C,'"K)  equals  the  sum  of  the  negative  coeffi- 
cients.    Compare  Exs.  3,  4,  and  6,  above. 

'68.  Binomial  factors  of  x"  ±a".  The  method  of  the  preceding 
article  may  be  used  to  find  binomial  factors  of  the  expressions 
x'^  —  a"  and  a;"  +  a**,  wherein  x  and  a  represent  any  numbers  what- 
ever, and  n  is  a  positive  integer. 

(i)  Thus  a;"  —  a"  is  exactly  divisible  by  x  —  a,  whatever  integer 
n  may  be,  because  if  a  be  substituted  for  x,  the  expression  a?"  —  a" 
becomes  a"  —  a",  i.e.,  0. 

Hence,  the  difference  of  lihe  positive  integral  powers  of 
two  numhers  is  exactly  divisible  hy  the  diff'erence  of  the 
nwmbers. 

By  actual  division,  it  is  found  that 

x  —  a  x  —  a  x  —  a 

^^  ~  "^  =  .t4  +  x^a  +  a;2a2  +  xa^  +  a* ;  etc. 
x  —  a 

Binomials  of  the  form  a:»  —  a"  can  always  be  separated  into  at  least  two 
factors,  both  of  which  may  be  written  down  by  inspection ;  one  of  these  fac- 
tors is  cc  —  a  and  the  other  is  cc«-i  +  x^-'^a  +  x^-^a^  -j 1-  a;a"-2  +  a^-i ;  this  last 

factor  is  homogeneous,  of  degree  n  —  1,  in  the  two  numbers,  and  contains  n 
terms,  all  of  which  are  positive. 


67-68]  FACTORING  103 

(ii)   Again,  x -\-  a,  i.e.,  x—  (—a),  is  a  factor  of  aj"  —  a"  when  n 
is  even,  because  then  (—  o)"  —  a'*  =  a"  —  a"  =  0  (§  18,  note  2). 

Hence,  ^7^e  difference  of  like  even  positive  powers  of  two 
numbers  is  exactly  divisible  by  the  sum  of  the  numbers. 

By  actual  division,  it  is  found  that 
*^ ~  *^ -  ° - ' •    ^^~^^  =  s^-sn  +  sP-t^',    ^lll^^s^-sH+sH^-sH^+st^-t^ ;  etc. 


s  +  t  s->rt  s+t 

The  student  may  make  a  verbal  statement  of  this  case  of  factoring  similar  to 
the  last  paragraph  in  (i)  above. 

(iii)  Again,  x-\-a,  i.e.,  »  —  (—  a),  is  a  factor  of  a;"  +  a"  when  n 
is  odd,  for  in  that  case  (—  a)"  +  a"  =  —  a"  +  a"  =  0  (§  18,  note  2). 

Hence,  the  suuv  of  like  odd  positive  powers  of  two  num- 
bers is  exactly  divisible  by  the  sum,  of  these  numbers. 

By  actual  division,  it  is  found  that 

ai±X^=a;2-a;?/  +  ?/2;        ^^^^^^  =  x^  -  x^y  +  x^^  -  xy^ -\- y^  • 
x+y  x+y 

2^i^  =  x6  —  x5y  +  a;4y2  _  x^yB  +  xhj^  —  xy^  +  ?/8 ;  etc. 
x-\-y 

The  student  may  formulate  this  principle  into  words,  —  see  last  paragraph 
in  (i)  above. 

(iv)  Finally,  a;  —  a  is  never  a  factor  of  a?"  +  a"  ;  for  if  a  be  sub- 
stituted for  X  in  this  expression  it  becomes  a"  +  a",  which  is  not 
0  either  when  n  is  even  or  when  n  is  odd,  and  therefore  x^  -f  a"  is 
not  exactly  divisible  by  a;  —  a  (§  48). 

Note.    Principles  (i)  to  (iv),  above,  may  be  briefly  recapitulated  thus: 
xn  —  an  is  always  divisible  by  a;  —  a, 
xn  —  a"  is  divisible  by  x  +  a  only  when  n  is  even, 
x"  +  a"  is  divisible  by  x  +  a  only  when  n  is  odd, 
x"  +  a«  is  never  divisible  by  x  —  a. 

EXERCISES 

/     1.  'Show  by  means  of  the  remainder  theorem  that  x^  —  a^  is  exactly 
divisible  by  a:  —  a ;  also  that  x^  +  a^  is  exactly  divisible  by  a:  +  a. 

2.  Prove  that  a;  —  a  is  a  factor  of  x"  —  a^  for  every  positive  integral 
value  of  n. 

3.  Prove  that  ar  +  a  is  a  factor  of  a:"  +  a"  for  odd  positive  integi-al 
values  of  n,  and  of  a;"  —  a"  for  even  positive  integral  values  of  n. 


104  ELEMENTARY  ALGEBRA  [Ch.  VII 

4.   Prove  that  neither  x  —  a  nor  x  +  a  is  a  factor  of  x"*  +  a"  when  n 
is  an  even  positive  integer. 

Write  out  the  following  quotients  by  inspection,  and  then  verify  them 
by  actual  division : 

21. 
22. 

23. 
24. 
25. 
26. 
27. 


5 

x'^-y\ 

x-y 

6. 

x^-y^ 

x-y 

7. 

a^-b* 

a-b 

8. 

w8  _  ^,B 

U  —  V 

9. 

v^-w^ 

V  -\-  w 

10. 

m^  —  n* 

m  +  n 

11. 

uS-yS 

w  +  u 

19 

x^  +  y^ 

13. 

a:5  +  2/5 
x  +  2/ 

14. 

m^  +  s^ 
m  +  s 

15. 

a^  +  b^ 
a-\-b 

16. 

(x^y  +  (^/2)^ 

x'^  +  y'^ 

17 

(2  a)4  -  a:* 

2a-x 

18. 

m6  -  32 

m-2 

19. 

4P-9 
2ifc-3 

9n 

16p*-81 

28. 


2:2  +  3,2 
81a^-16 

3a  +  2 
64 -rg 
r  +  2  ' 
27  a:8  +  64  gS 

3x  +  4a 
32x5  +  1 
2a:  +  l  ' 

x^  +  yg 
x2  +  3/2* 

a2  +  62  ' 
32  xio  +  ?/i6 


a; +  2/  2jt)  +  3  2  a;2  _|- yS 

29.  Compare  the  quotients  in  Exs.  5-15  with  the  corresponding 
powers  of  a  binomial  (§  62),  with  reference  to  coefficients,  exponents, 
signs,  etc. 

30.  Of  what  is  x^  the  square  ?     Of  what  is  it  the  cube  ? 
Write  x^  —  y^  as  the  difference  of  two  squares ;   of  two  cubes. 

Is  a;2  -  2/2  a  factor  of  x^  -  y^l    Why?    Is  x^  -  y^2    Is  x^  +  /?    Why? 
Find  the  prime  factors  of  x^  —  y^. 

31.  When  seeking  the  prime  factors  of  x^  -  y^  show  that  it  is  better 
not  to  divide  out  the  factor  x  —  y  at  once,  but  rather  to  separate  x^  —  y^ 
first  into  the  factors  x^  —  y^  and  x^  +  y^,  and  then  to  separate  each  of 
these  factors  further.  Is  a  similar  plan  advisable  in  general,  e.g.,  with 
a;8-y8  and  p"^  -  720? 

32.  Find  the  prime  factors  of  mi2  _  ni2 ;  compare  Ex.  31. 

33.  Find  the  prime  factors  of  x^  -  3/^ ;  also  of  64  a^  _  1. 

34.  Prove  that  jt)«  —  r"  is  exactly  divisible  by  p^  -  r2,  if  n  is  an  even 
positive  integer. 

35.  For  what  positive  integral  values  of  n  between  1  and  9  has  a;"  +  «/" 
no  binomial  factor  ?    Is  ic2  +  3/2  a  factor  of  x^ -\- y^t 


68-69]  FACTORING  105 

Resolve  the  following  expressions  into  their  prime  factors : 

36.  x*-y\  40.    aioxio-yio.  44.    S  as^^  -  S  at^^. 

37.  a^-b^  41.  p9  +  l.  45.   jfi  +  y^. 

38.  a8-68.  42.   16  a*^^  -  81  xV-  46.   64  x^  +  y«. 

39.  m8  -  1.  43.    ai2a:i3  _  b^^xy^-2, 

69.  Factoring  by  rearranging  and  grouping  terms.  A  rearrange- 
ment and  grouping  of  the  terms  of  an  expression  will  often 
reveal  a  factor  which  could  not  be  easily  seen  before. 

Ex.  1.   Find  the  factors  of  ax  —  3  &y  +  6x  —  3  ay. 

Solution,     ax  —  dby  +  bx  —  Say  =  ax  +  bx  —  Sby  —  day 

=  x(a  +  5)  -  3  y(a  +  b) 
=  (a  +  6)(x-3y). 

Ex.  2.  Find  the  factors  of  x(x  +  4)  -  ^(^^  +  4). 

Solution.      x(x  +  4)  —  y(y  +  4)  =x^-\-4:Z  —  y^  —  iy 

=  x-2~y^  +  4(x  -  y) 
=  (^ -y)(^ +  Z/ +  4). 

Note.  The  factor  z  —  y  could  also  have  been  detected  by  means  of  §  67, 
because  the  given  expression  is  zero  when  x  =  y. 

EXERCISES 
Find  the  factors  of  the  following  expressions  : 

3.  ax8  +  1  +  a  +  X.  7.   m^  -  n^  -  (m  -  n)^. 

4.  a2J2  +  a2  +  62  ^  1.  8.   x^  +  x2  _  4  a- _  4. 

5.  ac  ■\-  bd  —  ad  —  he.  9.   5  x^  +  1  —  x^  —  5  x. 

6.  ac"^  +  bd"^  -  ad^  -  bc^.  10.   a^  -  9  x2  +  4  c2  -  4  ac. 

Suggestion.  The  first,  third,  and  fourth  terms  of  the  expression  in  Ex.  10 
are  together  (a  — 2c)2,  i.e.,  the  given  expression  equals  (a  — 2c)2  — 9^2^  of 
which  the  factors  are  obvious. 

11.  X*  —  xy^  —  ax^  +  ay^-  15.   ab  +  bx"*  —  x''^™  —  ay^. 

12.  1  +bx-  (a^  +  ab)x^.  16.   3  xy(x  +  y)  +  16  x^  +  16  y^ 

13.  a^c^  +  acd  +  abc  +  bd.  17.    (p^  -  q^y  -  (p^  -  pgy. 

14.  x4  -  4  xY  +  2  x8  -  16  y^  18.    (x  +  yy  +  12  (x  +  y)  -  85. 

19.  a^x  4-  abx  +  ac  +  b^y  +  aby  +  be. 

20.  (x2  +  6  X  +  9)2  -  (x2  +  5  X  +  6)2. 

21.  x^+  (a  +  b  -  c)x2  +  (ab-ac  -  bc)x  -  abc. 


106  ELEMENTARY  ALGEBRA  [Ch.  VII 

22.  m^  +  n^  -f  m  +  77Jn  +  n  +  run. 

23.  14  a{x  -  ?/)  +  49  a2  +  (x  -  yy. 

24.  a;2  -  a2  +  y2  _  ^,2  4.  2  arz/  -  2  a6. 

25.  A2  _  „i2  +  10  m  +  F  -  25  -  2  ^^. 

26.  9  a2  +  12  a6  +  4  &2  _  15  a  -  10  &  -  24. 

27.  a2  +  &2  +  c2  +  2  (rt6  +  «c  +  &c)  +  5  (a  +  &  +  c). 

28.  a:2  +  3^2  +  ^2  +  2  (a:?/  +  X2:  +  3/z)  +  5  (a;  +  3/  +  z)  +  6.     ' 

29.  4  a;2  +  10  X  +  6  -  5  a  -  4  aa;  +  a2. 

70.  Factoring  by  means  of  other  devices.  It  often  happens  that 
the  factors  of  an  expression  will  become  apparent  by  adding  a 
certain  number  to,  and  subtracting  the  same  number  from,  the 
given  expression ;  this,  of  course,  leaves  the  value  of  the  expres- 
sion unchanged. 

Ex.  1.   Find  the  factors  of  a;*  +  a;2  +  1. 

Solution.     If  the  second  term  in  this  expression  were  2  x^  instead 
of  a;2,  then  [§  66  (i)]  the  expression  could  be  written  (x2  +  1)2;   this 
suggests  that  x^  be  both  added  and  subtracted,  which  gives 
a:*  +  a;2  +  1  =  x*  +  2  a;2  +  1  -  a;2 
=  (:c2  +  1)2  _  a-a 

=  (:c2  +  1  4.  a.)  (3,2  4.  1  _  a-)^  [■§  66  (ii) 

i.e.,  a;4  +  a:2  +  1  =  (a:2  +  x  +  1)  (a:2  -  a:  +  1). 

Ex.  2.    Find  the  factors  of  a"^  +  a^h"^  +  &*. 

Solution.  This  expression  may  be  treated  in  the  same  way  as  Ex.  1, 
*^^^  •  a*  +  a2j2  +  ft4  ^  ^4  .|.  2  a252  ^.  54  _  ^2^2 

=  (a2  +  &2')2  _  ^cihy 
=  (a2  -\-ah  +  62)  (^^2  -  ab  +  b^). 
Ex.  3.   Find  the  factors  of  a;^  -  4  a;  -  32. 

Solution.  Here  the  first  two  terms,  plus  4,  are  an  exact  square,  and 
this  suggests  the  following  arrangement : 

a;2  -  4  X  -  32  =  a;2  -  4  a;  +  4  -  32  -  4 

=  (a;  _  2)2  _  36 

=  (a:  _  2  +  6)  (a:  -  2  -  6), 

I.C.,  a;2  -  4  a:  -  32  =  (a:  +  4)(a;  -  8). 

Note.  Observe  the  superiority  of  the  method  of  Ex.  3  over  the  method  of  §  66 
(iii)  for  factoring  the  same  expressiou. 


69-70]  FACTORING  107 

EXERCISES 

Factor  the  following  expressions : 

4.  m^  +  m2n2  +  n*.  13.  5  a;*  -  70  xY  +  5  y*- 

5.  p^+4:  qK  14.  9  a*  +  26  a%'^  +  25  6*. 

6.  a:2  +  6  a;  +  5.  15.  a^  +  2  a6  -  rf^  _  2  6rf. 

7.  9  s2  +  30  s<  +  16  A  16.  x^  +  64  y^. 

8.  a;4  +  a2x2+a4.  17.  4a4+81. 

9.  a-8  +  a:4//4  +  ?/8.  18.  x^y^  +  4  ar^/*. 

10.  4  a8  -  21  a4&4  +  9  68.  19.  m6  +  4mn4. 

11.  a4j4  +  ^262^2^2  ^  ^4^4.  20.  a^  +  8  a2  _  128. 

12.  9  x^  +  8  a:22/2  +  4  3/*.  "  21.  5  na;*  -  70  na;2  +  200  n. 

22.  What  must  be  added  to  a:*  +  3  x2  +  4  to  make  it  an  exact  square? 
What  must  then  be  subtracted  to  leave  the  result  unchanged?  Factor 
this  expression. 

23.  What  must  be  added  to  a;*—  3  x2  +  4  to  make  it  an  exact  square,  and 
what  must  then  be  subtracted  so  as  not  to  change  the  value  of  the  given 
expression  ?  If  the  given  expression  is  written  in  the  form  (a:2  —  2)2  +  a:2, 
can  it  be  factored  by  any  of  the  preceding  methods  (cf.  §  68)  ? 

24.  Can  the  sum  of  two  squares  be  factored  (cf.  §  68)  ?  Is  not  the 
expression  in  Ex.  5  above  the  sum  of  two  squares?  Could  this  expres- 
sion be  written  (/)2  +  2  ^2)2  _  (2^9^)2? 

25.  Factor  the  expression  3  a:2  +  z  -  10  [cf.  Ex.  1,  §  66  (iv)]. 

Solution.  Zx^  +  x-\0  =  12(3a;2  +  cc-10) 

12 

^36a;2  +  ]2a;-120 
12 
36x2  + 12  a; +  1  —  121 
12 
_  (6  a; +  1)2 -(11)2 

12 
_  (6  a; +  12)  (6  a; -10) 

12 
=  (a;  +  2)(3x-5). 

Note.  The  above  method  is  more  direct  than  that  given  in  §  66  (iv)  ;  it  con- 
sists in  multiplying  the  given  expression  by  such  a  number  as  will  make  its 
highest  term  an  exact  square,  and  the  next  highest  term  exactly  divisible  by  tivice 
the  square  root  of  the  highest  term,  then  factoring  the  resulting  expression  as 
explained  in  §  70,  and  finally  dividing  the  whole  by  the  number  first  used  as  a 
multiplier,  so  as  not  to  change  the  value  of  the  expression. 


108  ELEMENTARY  ALGEBRA  [Ch.VII 

Factor  the  following  expressions  : 


26. 

5  ,n2  -  2  m  -  3. 

30. 

8.12  +  23^5-  352. 

27. 

6«2_n«_35. 

31. 

4iV2+16iViM3  +  15  M6. 

28. 

18a:2-  3  a; -36. 

32. 

2  a;2  +  5  ar«/  +  2  ?/2. 

29. 

6  i?2  _  2  72  _  20. 

33. 

3  a:2  -  10  x?/  +  3  ?/2. 

71.  General  plan  for  factoring  a  polynomial.  Based  upon  §  §  65-70, 
the  following  suggestions  for  separating  a  polynomial  into  its 
prime  factors  may  be  made,  ^j  inspection  find  the  monomial 
factors  of  the  given  polynomial,  if  there  are  any  such,  and  then 
write  this  polynomial  as  the  indicated  product  of  the  monomial 
and  the  corresponding  polynomial  factor ;  then,  by  rearrangement 
of  the  terms,  or  by  some  one  of  the  other  methods  given  above, 
separate  this  polynomial  factor  into  two  factors,  and  replace  it 
by  their  indicated  product;  then  further  separate  each  of  these 
factors  into  two  others,  if  possible,  and  so  continue  until  all  of 
the  factors  are  prime. 

EXERCISES 

Factor  the  following  expressions : 

1.  m^x^  +  7n2^5.  4.  x^  ■\-  ax  —  ay  —  yx. 

2.  c2  -  5  c  -  14.  5.  a;4  -  8  a:3  +  15  x^. 

3.  21  m^-ma-  10  a\  6.  m%4  -  5  m'^n'^  +  4. 

7.  25  a2  +  2/2  +  10  a;2  +  10  a?/  -  35  aa;  -  7  xy. 

8.  7n2  +  6  m  -  a:2  +  9  -  4  x?/  -  4  y2^ 

9.  2  (a2&2  _  a2c2  +  52^.2)   _   ^^^4  _^  J4  _^  ^4). 

10.  ari2  _  2,12.  16.  ai6  +  1. 

11.  ai2a:i22^i2  +  ri2si2.  17.  (a2  +  5a  +  4)2-(a2_5a-6)2. 

12.  4  ax2  4-  4  ay\  18.  a;2«  2  +  ^,2^2  _^  2  a;»-%. 

13.  a%H'^  +  4  ah'^xy  +  4  h'^y\  19.  x^  -  y^  -  ^  x^  +  3  a:22/4. 

14.  32  a  -  ax^.  20.  a;^!/  -  15  x'^y  +  38  xy  -  24  ?/. 

15.  a9  +  4  a.  21.  js  4.  ^,4^,2  ^  ^4. 

22.  w%3  4.  2  mhi''rh^  +  m%iV%«. 

23.  a;2  +  9  y2  +  25^2  _  6  a:?/  -  10  xs  +  30  3^2. 

24.  a:5  +  5  x^az^  +10  a;^^^*  +  10  0:20826  4.  5  xaH^  +  a^sio. 

25.  a2  -  2  a6  +  &2  _  2  ac  +  2  6c  +  c2  -  2  a(/  +  2  Jrf  +  2  cd/  +  tf^. 


70-72]  FACTORING  109 

72.  Solving  equations  by  factoring.  If  all  the  terms  of  an 
equation  be  transposed  to  its  first  member,  factoring  that  member 
will  always  simplify  the  finding  of  the  roots  of  the  given  equa- 
tion ;  this  is  illustrated  by  the  following  examples. 

Ex.  1.  Given  x'^  —  5  x  +  Q  =  0;  to  find  its  roots,  i.e.,  to  find  those 
values  of  x  for  which  this  equation  is  satisfied  (cf.  §  23). 

Solution.  By  §  66  (iii)  the  first  member  of  this  equation  is  the  prod- 
uct of  a;  —  3  and  x  —  2,  and  the  given  equation  may,  therefore,  be  written 

(x-2)(x-  3)  =0. 

It  is  manifest,  moreover,  that  a  product  is  0  if,  and  only  if,  at  least  one 
of  its  factors  is  0 ;  hence  (x  —  2)(x—  3)  =  0  if,  and  only  if, 

a:-2  =  0ora;-3  =  0, 

i.e.,  if,  and  only  if,  a:  =  2  or  re  =  3  ; 

hence  the  roots  of  the  given  equation  are  2  and  3. 

Ex.  2.    Given  x^  =  ^  x  +  i;  to  find  its  roots. 

Solution.     On  transposing,  this  equation  becomes 
a;2  _  3  a;  -  4  =  0, 
i.e.,      -,  (x  -  4)  (a:  +  1)  =  0 ;  [§  66  (iii) 

hence  either  a;  —  4  =  0  or  x  -f  1  =  0,  i.e.,  a;  =  4  or  a:  =  —  1, 

and  therefore  the  roots  are  4  and  —  1 . 

Ex.  3.    Solve  the  equation  6  x^  —  11  a;  =  35. 

Solution.     Transposing  and  factoring  [§  66  (iv)],  this  equation  may 

be  written 

(3x+5)(2x-7)  =  0; 

hence  3a;  +  5=:0  or  2x  —  7  =  0,  i.e.,  x  =  —  f  or  x  =  |, 

and  therefore  the  roots  are  —  |  and  |. 

Note.  Since  the  roots  of  the  equation  {x  —  a)(x—b)—0  are  a  and  6,  therefore 
an  equation  which  shall  have  any  given  numbers  as  roots  may  be  immediately 
written  down ;  thus  the  equation  whose  roots  are  3  and  8  is 

(x -3){x-8)=  0,  i.e.,  cc2— 11  a;  +  24  =  0. 

Similarly,  the  equation  whose  roots  are  2,-1,  and  5  is 

(a;  -  2)  (a;  + 1)  (x  —  5)  =  0,  i.e.,  a:8  —  6  a;2  +  3  a;  + 10  =  0. 


110  ELEMENTARY  ALGEBRA  [Ch.  VII 

EXERCISES 

4.  What  is  meant  by  a  root  of  an  equation  ?  May  an  equation  have 
more  than  one  root? 

5.  Find  the  roots  of  x^  —  4  a:  —  21  =  0.  Verify  their  correctness  by 
substituting  them,  in  turn,  for  x  in  the  given  equation. 

6.  Solve  the  equation  ^^  _  g  ^  ^  5  —  o,  and  verify  the  solution. 

7.  What  values  of  x  will  satisfy  the  equation  (a;  —  2) (x  —  3)  =  0?  If 
X  =fz  2*  will  a:  -  2  be  0  ?    If  a:  ^  3,  will  ar  -  3  be  0  ?    If,  then,  x  is  neither 

2  nor  3,  can  the  given  equation  be  satisfied  ?     This  equation  has  then 
how  many  roots? 

8.  Write  the  equation  whose  roots  are  5  and  1.  Also  one  whose  roots 
are  3,  2,  and  7. 

9.  W^rite  the  equation  whose  roots  are :  1  and  —  5 ;  |  and  6  ;  a  and  6 ; 
3,  —  1,  and  5 ;  a,  —  a,  and  2a;  1,  2,  3,  and  4. 

Solve  the  following  equations,  and  verify  the  correctness  in  each  case : 

10.  a:2-2a:  =  15.         13.    Sy^  +  15  =  -  26  3/.     16.  2a:3  +  5a:2=  2  a;  +  5. 

11.  6  a;2  _  a:  -  1  =  0.    14.   5  a;^  -  7  a;  =  0.  17.  a:^  -  4  =  0. 

12.  ^if+y=  10.         15.    12  22  =  4  2.  18.  x^  -  13  a:2  +  36  =  0. 
19.   x3  +  a;2-x  =  l.                           20.    (a:  -  l)(a;+ l)(a:  -  2)  =  0. 

21.  Can  the  roots  of  the  equation  in  Ex.  20  be  determined  by  mere 
inspection  ?    Can  the  roots  of  the  equation 

(3a:-2)(a;+l)=2 

be  so  determined  ?    What  are  these  roots  ? 

22.  Write  out  a  rule  for  solving  such  equations  as  those  given  in  the 
above  examples. 

PROBLEMS 
By  the  meth6d  of  §  26  f  solve  the  following  problems : 

1.  If  the  product  of  the  two  remainders  obtained  by  first  subtracting 

3  from  a  certain  number,  and  then  5  from  the  same  number,  is  24,  what 
is  that  number?     How  many  solutions  has  this  problem?     Explain. 

2.  If  the  sum  of  two  numbers  is  12  and  one  of  these  numbers  is  x, 
what  is  the  other  number?  Find  two  numbers  whose  sum  is  12  and  of 
which  the  square  of  the  larger  is  1  less  than  10  times  the  smaller. 

*  The  expression  a;  ::^2  is  read  "  x  is  not  equal  to  2." 
t  §  26  should  now  be  re-read. 


72]  FACTORING  111 

3.  The  difference  between  two  numbers  is  2,  and  the  sum  of  their 
squares  is  130.     What  are  these  numbers? 

4.  One  side  of  a  rectangle  is  3  feet  longer  than  the  other.  If  the 
longer  side  be  diminished  by  1  foot  and  the  shorter  side  increased  by 
1  foot,  the  area  of  the  rectangle  will  then  be  30  square  feet.  How  long 
is  this  rectangle  ? 

5.  A  rectangular  orchard  contains  2800  trees,  and  the  number  of  trees 
in  a  row  is  10  less  than  twice  the  number  of  rows.  How  many  trees  are 
there  in  a  row? 

6.  If  the  dimensions  of  a  certain  rectangular  box,  which  contains 
120  cubic  inches,  were  increased  by  2,  3,  and  4  inches,  respectively,  the 
new  box  would  be  cubical  in  form.     Find  the  dimensions  of  this  box. 

7.  Plow  may  $128  be  divided  equally  among  a  certain  number  of 
persons  so  that  the  number  of  dollars  received  by  each  person  shall 
exceed  the  number  of  persons  by  8  ? 

8.  A  certain  club  banquet  is  to  cost  $75,  and  it  is  found  that  this 
will  require  each  member  of  the  club  to  pay  50  cents  more  than  ^q  as 
many  dollars  as  there  are  members  in  the  club.  How  much  must  each 
pay,  and  how  many  members  are  there  in  the  club  ? 


CHAPTER   VIII 

HIGHEST  COMMON  FACTORS  —  LOWEST  COMMON  MULTIPLES 

I.     HIGHEST  COMMON  FACTORS 

73.  Definitions.  A  factor  of  each  of  two  or  more  numbers  or 
algebraic  expressions  is  called  a  common  factor  of  these  numbers 
or  expressions;  the  highest  common  factor  —  usually  designated 
by  the  letters  H.  C.  F.  —  of  two  or  more  numbers  or  expressions  is 
the  product  of  all  the  prime  factors  (§  63)  that  are  common  to 
these  numbers  or  expressions. 

E.g.,  the  H.  C.  F.  of  12  a^h'^cx^  and  6  ah^x.^y  is  6  ah^x'^,  because  when  this  factor 
is  removed  from  the  given  expressions  they  have  no  comrnon  factor  left ;  6  ab^^ 
is  then  the  product  of  all  the  common  prime  factors  of  the  given  expressions. 

Similarly,  3a(a;-l)2(a;  — 2)  is  the  H.C.F.  of  6  a2x(x  — l)4(a;  — 2)(a  — ?/)  and 
15  ab{x  —  y){x  —  l)2(a;  —  2)3. 

Note.  It  is  evident  from  the  above  definition  that  no  common  factor  of  two 
or  more  expressions  is  of  higher  degree  in  any  letter  than  their  H.  C.  F. 

Two  or  more  numbers  or  algebraic  expressions  which  have  no 
common  factor  except  unity  are  said  to  be  prime  to  each  other. 

74.  Highest  common  factor  of  two  or  more  monomials.  From 
the  definition  and  illustration  given  above,  it  is  clear  that  the 
H.  C.  ¥.  of  two  or  more  monomials  can  be  found  by  inspection. 

E.g.,  to  find  the  H.  C.  F.  of  12  a%^xy,  6  ab^z^,  and  9  ab^^. 

Inspection  shows  that  these  monomials  have  the  prime  factors  3,  a,  b,  b,  and  x 
in  common,  and  that,  when  these  are  removed,  there  are  no  other  factors  common 
to  the  given  monomials ;  hence  their  H.  C.  F.  is  3  •  a  •  6  •  6  •  x,  i.e.,  3  ab^x. 

A  rule  for  writing  down  the  H.  C.  F.  of  several  monomial  ex- 
pressions may  be  formulated  thus:  to  the  H.C.F.  of  the  nu- 
merical coefficients  annex  those  letters  that  are  found  in 
each  one  of  the  given  monomials,  and  give  to  each  of  these 
letters  the  lowest  exponent  which  it  has  in  any  of  the 
monomials. 

112 


73-75]  HIGWEST  COMMON  FACTORS  113 

EXERCISES 

Find  the  H.  C.  F.  of  the  following  sets  of  monomials: 

1.  3  a2j8c(^  and  6  a6  Vrf3. 

2.  15  x^z,  24  xYz\  and  18  x^. 

3.  16  x'^yHhn%  169  y^z^,  and  39  x'^y^m*. 

4.  2041  a^i^cT  and  8476  a%c^d. 

5.  292  x^y'^z^,  1022  x^^^^  and  1095  x^^*. 

6.  364  x-'^if'^z^  and  455  x'^y'^^'z^. 

7.  Is  the  H.  C.  F.,  as  above  defined,  the  same  as  the  greatest  common 
divisor  (G.  C.  D.)  in  the  arithmetical  sense?  What  is  the  H.  C.  F.  of  aH^y 
and  a^xy^'i    Is  this  H.  C.  F  also  the  G.  C.  D.  when  a  =  ^,x=Q,  and  y  —  ^'i 

Note.  Observe  that  highest  refers  to  degree,  while  greatest  refers  to  value. 
If  c  is  any  proper  fraction,  then  c  >  c^  >  c^  •  ••,  but  c"  is  always  higher  than  c^. 

Find  the  H.  C.  F.  of  the  following  sets  of  expressions  : 

8.  24  aH{y  -  zy{w  +  3)  and  56  a%x^{y  -  zy(w  +  3)2. 

9.  473  hhH(^x  -  1)2(3  -  2  yy  and  319  a^hs\x  -  l)(x  -  2)2(3  -  2  yy. 

75.  H.  C.  F.  of  two  or  more  polynomials  whose  prime  factors  are 
known.  The  H.  C.  F.  of  several  polynomials  whose  prime  factors 
are  known  may  be  written  down  by  inspection  as  is  done  for 
monomials  in  §  74. 

EXERCISES 

Find  the  H.  C.  F.  of  each  of  the  following  sets  of  expressions: 

1.  4(rt  +  J)3(a-J)  and  J(a  +  &)2(a  -  &)2. 

2.  Q(a  +  by{a-hy  and  15(a- &)2(a  +  &). 

3.  4  aa;2  -  20  ax  +  24  a  and  6  06^2  +  24  ahx  -  126  ah. 

Solution 

Since  4 a3:2- 20  ax  +  24 a  =  4 «(a;2- 5 a;  +  6)  =  2  •  2  a(x-  2)  (x - 3), 
and  6  a&x2  +  24  ahx  - 126  a6  =  6  a6  (a;2  +  4  a;  -  21)  =  3  . 2  a&  (x  +  7)  (a;  -  3), 
therefore  the  H.  C.  F.  is  2  a  (a;  —  3) . 

4.  a2-&2,  a{a  +  h),  and  a2  +  2a&  +  62. 

5.  5-19iP-4a:2  and  2^2 +  7  a: -15. 


114  ELEMENTARY  ALGEBRA  [Ch.  VIII 

6.  z2  +  5  :c  +  6,  x2  +  7  a;  +  10,  and  x^  +  12  a;  +  20. 

7.  a^-a-12   and  a^  -  4  a  -  21. 

8.  15(^2  -  z)  and  35(?/%  -  ?/2;). 

9.  X*  +  a:^^'^  +  ?/''  and  (x^  -  a:?/  +  ?/2)2, 

10.  Of  what  is  the  H.  C.  F.  of  two  or  more  expressions  composed? 
State  a  rule  for  finding  the  H.  C.  F.  of  two  or  more  expressions  which 
are  already  separated  into  their  prime  factors,  or  which  may  be  easily  so 
separated. 

11.  What  is  the  H.  C.  F.  of  x^x-iy  and  a:(a;2-l)?  Is  this  also 
the  G.  C.  D.  of  these  expressions  for  all  values  of  x  V  Try  05  =  3,  and  also 
x  =  i.     Compare  Ex.  7,  §  74. 

Find  the  H.  C.  F.  of  the  following  sets  of  expressions: 

12.  4  afeV  -f  12  ab^x  -  40  ah\    6  aH'^y  -  6  a^xy  -  12  o?-y,    and 

18  a%x2  -  54  a'^mx  +  36  ahn. 

13.  15  a4a;2  +  15  a%H^  +  15  }fix'^  and  3(a2  _  aW-  +  h% 

14.  a:8  +  aS  and  3  a^  +  3  a^  -  5  ax^  -  5  x\ 

15.  2  a;2  -  X  -  3  and  2  x^  +  11  a;^  -  a:  -  30.* 

16.  (a;+3)(a:2-4),  x^H- 4a:3  + 2  a;2  -  x+ 6,   and  2  a:8  + 9  a;2+ 7  x  -  6. 

17.  a3  +  l,   3a3-4a2  +  4a-l,   and   2a3+a2-a  +  3. 

76.  H.  C.  F.  of  two  polynomials  neither  of  which  can  be  readily 
factored.  Although,  it  is  only  in  exceptional  cases  that  the  factors 
of  a  polynomial  can  be  found  (such  cases  were  examined  in  Chap- 
ter VII),  yet  the  common  factors  of  any  two  given  polynomials 
can  always  be  found. 

The  method  for  finding  the  H.  C.  F.  of  two  polynomials  neither 
of  which  can  be  readily  factored,  is  precisely  the  same  as  that 
used  in  arithmetic  for  finding  the  G.  C.  D.  of  two  numbers,  neither 
of  which  can  be  easily  factored. 

*  Since  the  second  of  these  expressions  is  not  easily  factored,  —  although  the 
first  is,  —  find  by  trial  whether  the  factors  of  the  first  expression  are  also  factors 
of  the  second. 

This  method  may  be  employed  whenever  any  one  of  a  given  set  of  expressions 
is  easily  separated  into  its  prime  factors. 


76-76] 


HIGHEST  COMMON  FACTORS 


115 


To  illustrate,  let  it  be  required  to  find  the  G.  C.  D.  of  1183  and  2639. 
1183)2039(2 
2366 
273)1183(4 
1092 
91)273(3 
273 
0 
The  last  divisor,  91,  is  the  G.  C.  D.  of  the  given  numbers.    This  work  may  be 
more  compactly  arranged  thus : 

Quotients 


1183 


1092 
91 


2639 

2366 

273 

273 

0 


Similarly,  the  H.  C.  F.  of  x4_{_3x3  +  222_3a;_3  and  x^  +  x^  —  2  may  be 
found  thus: 

Quotients 


a*  +  3  a;3  + 


2x2- 


3a;- 

2cc 


2x3  +  2x2—    X  — 3 
2  xs  +  2  x2  —  4 


x  +  1 


x  +  2 


x2-2x-2 


X8  +  X2  ■ 
X3  — X2 


2x2- 
2x2- 


2 
2x 


2x-2 
2x-2 


and  —  x  +  1,  which  is  the  last  divisor,  is  the  H.C.F.  of  the  given  polynomials.* 

The  procedure  illustrated  above  may  be  formulated  in  words 
thus: 

Arrange  the  given  polynomials  according  to  the  descend- 
ing powers  of  some  common  letter,  and  divide  the  higher 
expression  hy  the  lower,  continuing  the  division  until  the 
remainder  is  of  lower  degree  than  the  divisor;  then  using 
this  rejnainder  as  a  divisor,  with  the  preceding  divisor  as 
a  dividend  {and  with  the  same  letter  of  arrangement), 
divide  as  before;  continue  this  process  until  the  remainder 
is  either  zero,  or  free  from  the  letter  of  arrangement :  —  if 
'  it  is  zero,  the  last  divisor  is  the  H.  C.  F.  sought ;  and  (cf .  §  77) 
if  it  is  free  from  the  letter  of  arrangement,  the  given  ex- 
pressions have  no  common  factor  containing  that  letter. 


*  The  H.  C.  F.  of  these  polynomials  may  also  be  regarded  as  x  —  1.    Why  ? 


116  ELEMENTARY  ALGEBRA  [Ch.  VIII 

EXERCISES 

By  the  above  method,  find  the  H.  C.  F.  of  the  following  pairs  of 
expressions : 

1.  a:2  +  5  a:  +  6  and  4  a;3  +  21  a:2  4-  30  ar  +  8. 

2.  12  a;4  -  8  a;3  -  55  a;2  -  2  a;  +  5  and  6  a;^  -  a;^  -  29  x  -  15. 

3.  6  a2  -  13  a  -  5  and  18  a^  -  51  a^  +  13  a  +  5. 

4.  5n4-10n3+lln2-6n+l  and  10n5_5n4_7w34-19n2-14n  +  2. 

77.  Fundamental  principle.  The  success  of  the  method  em- 
ploj^ed  in  §  76  for  finding  the  H.  C.  F.,  whether  in  arithmetic 
or  algebra,  is  due  to  the  following  principle: 

//  an  integral  algebraic  expression  *  he  divided  hy  another 
such  expression  which  is  of  the  same  or  of  a  lower  degree 
in  the  letter  of  arrangement,  and  if  there  he  a  remainder, 
then  the  H.  C.  F.  of  this  remainder  and  the  divisor  is  also 
the  H.  C.  F.  of  the  two  given  expressions. 

To  prove  the  correctness  of  this  principle,  let  Ei  and  E2  repre- 
sent any  two  given  integral  expressions,  and  let  the  degree  of  E2, 
in  the  letter  of  arrangement,  be  at  least  as  low  as  that  of  Ei ;  also 
let  Qi  and  Ri  represent,  respectively,  the  quotient  and  remainder 
when  El  is  divided  by  E2 ;  then  (§  47,  Ex.  11), 

E,  =  q,E2  +  Bi,  (1) 

whence,  Ri=  E^  —  q^E^.  (2) 

Now  since  any  factor  of  each  term  of  an  expression  is  a  factor 
of  the  whole  expression,  therefore  any  factor  common  to  E2  and 
Ri  is  also  a  factor  of  gi^/a  +  R\,  and  therefore,  by  equation  (1), 
of  Ex ;  i.e.,  all  the  factors  common  to  R^  and  E2  are  also  factors  of 
jEJi,  and  therefore  common  to  E2  and  Ey 

But,  by  exactly  the  same  reasoning,  equation  (2)  shows  that  all 
the  factors  common  to  E^  and  E^  are  also  common  to  E.^  and  R^ ; 

*  "  Integral  expression  "  as  here  used  includes  arithmetical  numbers  also. 


76-77] 


HIGHEST  COMMON  FACTORS 


117 


i.e.,  the  factors  common  to  Ri  and  E2  are  precisely  those  which  are 
common  to  E^  and  Eo.  Hence  the  H.  C.  F.  of  R^  and  E2  is  also 
the  H.  C.  F.  of  El  and  E^. 

From  the  proof  just  given  it  follows :  (1)  that  if  E2  be  now 
divided  by  R^,  giving  a  remainder  R2,  then  the  H.  C.  F.  of  R2  and 
i?i  is  also  the  H.  C.  F.  of  ^2  and  R^,  and  therefore  of  E^  and  E2. 
So,  too,  if  J?i  be  divided  by  R2,  giving  a  remainder  R^,  then  the 
H.  C.  F.  of  R2  and  R^  is  also  the  H.  C.  F.  of  E^  and  E^,  and  so 
on ;  i.e.,  the  H.  C.  F.  of  E^  and  E2  is  also  the  H.  C.  F.  of  any  two 
consecutive  remainders  in  this  succession  of  divisions. 

But  these  successive  remainders  are  of  lower  and  lower 
degrees,*  hence  a  remainder  i?„  which  is  either  0,  or  free  from 
the  letter  of  arrangement,  must  finally  be  reached ;  if  i?„  =  0,  then 
Rn-i  is  the  H.  C.  F.  of  i?„_i  and  i?n_2,  and  therefore  of  E^  and 
E2,  but  if  Rn  is  merely  free  from  the  letter  of  arrangement,  then 
Rn-\  and  R^-^  can  have  no  common  factor  containing  this  letter, 
and  therefore  E^  and  E2  have  no  common  factor  which  contains 
that  letter. 


Note.  It  follows  directly  from  the  definition  (§  73)  that  the  H.  C.  F.  of  two 
entire  expressions  is  not  altered  by  multiplying  or  dividing  either  of  them  by  any 
number  which  is  not  a  factor  of  the  other.  By  introducing  and  suppressing 
suitable  factors  during  the  divisions  above  described,  fractional  coefficients, 
which  might  otherwise  arise,  may  always  be  avoided. 

To  illustrate,  let  it  be  required  to  find  the  H.C.F.  of  3x3  +  8x2  +  3x  — 2 
and  a3_2a;24-a;  +  4. 

Since  these  expressions  are  of  the  same  degree,  either  one  may  be  used  as 
divisor;  the  work  may  be  arranged  thus: 

Before  beginning 
the  second  division 
the  factor  14  is  sup- 
pressed (see  note 
\  above),  and  later  2 
is  also  suppressed; 
fractional  coeffi- 
cients are  thus 
avoided. 


3a;8  +  8x2  +  3x-  2 
3a;3-6a;2  +  3x  +  12 

3 

x-2 
x-1 

x8-2x2  +  x  +  4 

X3                 -X 

14)14x2-14 
x2-l 

-2x2  +  2x  +  4 
-2x2           +2 

X2  +  X 

2x  +  2(2 

-x-1 
-x-1 

x  +  1 

0 

and  x  +  1,  which  is  the  last  divisor,  is  the  H.  C.  F.  of  the  given  expressions. 


*  If  El  and  E^  represent  arithmetical  numbers,  then  i?i,  R.2,  and  R2, 
sent  smaller  and  smaller  numbers. 


repre- 


118 


ELEMENTARY  ALGEBRA 


[Ch.  VIII 


As  a  further  illustration,  let  us  find  the  H.  C.  F.  of 

a;4  +  4a;3  +  2a;2  — x  +  6  and  2a;8  +  9a;2  +  7a;  — 6. 

Before  beginning 
the  division  the  fac- 
tor 2  is  introduced 
so  as  to  avoid  frac- 
tional coefficients  in 
the  quotient  (cf .  note 
above) ;  later  —  2  is 
introduced  for  the 
same  purpose;  and 
finally —3 is  rejected. 

cb2  +  5  a;  -f  6,  which  is  the  last  divisor,  is  the  H.  C.  F.  of  the  given  expressions. 


2 

2a;-l 

2a:3  +  9x2  +  7a;-6 
2a:3  +  10a;2  +  12a; 

2a;44-8a;8  +  4a;2-2x  +  12 

—  a;2  —  5  a;  —  6 

—  a;2  — 5a;  — 6 

-a;3-3a;2  +  4a;  +  12 
-2 

0 

2a;8  +  6a;2-8a;-24 
2a:3  +  9:r2  +  7a!-    6 

-3)-3a;2-15a;-18 

x^  +  5x  +  6 

EXERCISES 

By  the  above  method  find  the  H. CF.  of  the  following  pairs  of 
expressions  : 

1.  x^-dx^+dx-1   and   x* -2  x3  + 2  a:2 -2  a:  + 1. 

2.  8  x3  -  22  a:2  -F  17  a;  -  3   and   6  a;8  -  17  a;^  +  14  x  -  3. 

3.  a:6-4x4-f  5a:3-3a:2  +  3a:-2   and  2  x^  -  o  x^- +  x  +  2. 

4.  a:6-4a;4  +  5a:2_2   and   Sx^  +  5x  +  2. 

5.  a;5_  2x4 -2x8-11x2 -a:- 15  and  2  x5-7  xH4x8-15x2+x-10. 

6.  x6  +  x4  +  x8-x-2   and   3x6  +  x6-x2-l. 

7.  a8  +  3a2-2a-6   and  4a2- a  +  0^+ 4a4  -  12-}- 4a8. 

8.  1  -  4  m8  -f  3  m*   and   1  -  5  w3  -f  4  m*  -h  m  -  tw^. 

9.  x5-3x4-3x8-15-19x   and   3  x^  -  3  x^ -K  x^  -  15 -f  9  x2- x. 

10.  What  is  meant  by  the  H.C.F.  of  two  expressions  E^  and  iJg^ 
If  a  is  not  a  factor  of  E^,  how  does  the  H.  C.  F.  of  jEJ^  and  a  •  E^  compare 
with  the  H.  C.  F.  of  E^  and  E^f    Why?    Compare  §  77,  note. 

11.  If  a  is  a  factor  of  E^,  but  not  of  E2,  how  does  the  H.  C.  F.  of  E^ 
and  a  •  E^  compare  with  the  H.  C.  F.  of  E^  and  JEJg?  In  introducing  and 
suppressing  factors  during  the  process  of  division  (§  77),  what  special 
precaution  must  be  exercised,  and  why  ? 

12.  Suppose  that,  at  some  stage  of  the  work  in  an  exercise  like  those 
above,  the  divisor  is  2  x2  —  4  x  -{-  2,  and  the  dividend  x8  —  3x2  +  3x-fl; 
what  would  be  the  eifect  on  the  final  result  if  the  factor  2  were  intro- 
duced into  the  dividend  to  avoid  fractional  coefficients?  What  should 
be  done  in  this  case  instead  of  introducing  the  factor  2 ?     Why? 


77-78]  HIGHEST  COMMON  FACTOBS  119 

13.  Show  that  every  factor  common  to  A  and  B  is  also  common  to 
A  —  B  and  A  +  B;  and  also  to  7nA  +  nB  and  mA  —  nB.  Is  the  H.  C.  F. 
of  A  and  B  necessarily  the  H.  C.  F.  of  ^  -  5  and  ^  +  -B? 

78.  Supplementary  to  §§  76  and  77.  (i)  H.C.F.  of  poly- 
nomials which  contain  monoTnial  factors.  The  problem  of 
finding  the  H.  C.  F.  of  a  pair  of  polynomials,  either  of  which 
contains  monomial  factors,  is  usually  much  simplified  by  setting 
aside  these  monomial  factors  before  the  division  process  is  begun. 
Factors  which  are  common  to  the  given  polynomials  must,  of 
course,  be  reserved  as  factors  of  their  H.  C.  F. ;  all  others  may 
be  rejected. 

Thus,  to  find  the  H.  C.  F.  of 

6x5  +  18a;4  +  12x3-18a;2— 18x  and  Sax^  +  3ax^  —  6ax, 

remove  the  monomial  factors  Hx  and  3  ax  from  the  given  expressions,  and  the 
remaining  polynomial  factors  are,  respectively,  x^-{-3x^-{-2x^  —  Sx  —  S  and 
a;8-f-a;2_2  ;  the  H.C.F.  of  the  monomial  factors  is  Sx,  and  the  H.C.F.  of  the 
polynomial  factors  is  a;  —  1  (see  illustrative  example,  §  76) ;  hence  the  H.  C.  F.  of 
the  given  polynomials  is  3  x(a;  —  1). 

(ii)  H.  C.  F.  of  polynomials  zvhich  involve  several  letters. 
Although  the  examples  given  in  §  77  involve  only  one  letter,  yet 
it  should  be  especially  observed  that  the  demonstration  there 
given  applies  to  expressions  involving  any  number  of  letters. 

Thus,  if  the  given  expressions  involve  several  letters,  then,  to  find  whether 
they  have  a  common  factor  containing  any  particular  one  of  these  letters,  they 
need  only  be  arranged  according  to  the  descending  powers  of  that  letter,  and 
divided  as  above  described.  If,  therefore,  the  given  expressions  be  successively 
arranged  according  to  each  of  the  several  letters  which  they  have  in  common, 
and  divided  as  above,  then  all  their  common  factors  {i.e.,  their  H.C.F.)  will  be 
found. 

Manifestly,  however,  any  common  factor  which  contains  two  or  more  letters 
will  be  found  when  the  given  expressions  are  arranged  according  to  any  one  of 
these  letters. 

(iii)  H.  C.  F.  of  three  or  more  polynomials.  Since  the 
H.  C.  F.  of  three  polynomials  is  a  factor  of  each  of  them,  it  is  also 
a  factor  of  the  H.  C.  F.  of  any  two  of  them ;  therefore  the  H.  C.  Y. 
of  three  polynomials  is  found  by  first  finding  the  H.  C.  F.  of  any 
two  of  them,  and  then  the  H.  C.  F.  of  that  result  and  the  third 
polynomial.  By  continuing  this  process  the  H.  C.  F.  of  any  num- 
ber of  polynomials  may  be  found. 


120  ELEMENTARY  ALGEBRA  [Ch.  VIII 

EXERCISES 
Find  the  H.  C.  F.  of  : 

1.  21  ax  —  17  ax^  —  5  ax^  +  ax^  and  5  ax^  —  34  ax^  —  1  ax. 

2.  7  m^^pS  —  49  in^x  +  42  m^  and  14  ahnx^  + 14  a^mx^—  56  a%a:  —  56  a'hn. 

3.  48s3te4-162s3fxH54s3^  and  18  s^^a^^- 9  s%%a;- 48  s2f%a:2+ 2452^2^^x3. 

4.  6  ca:3(l  +  2/2)  -  18  cx^z  +  2  cy^  -  4:  cyH  +  12  cz^  -  2  C2(3  z/+  2)  +  2  cy 
and  2  a?/4  +  2  ax2(3/2  _  3  ;2)  _  6  03/2^  +  2  a(a:2  _  2/2')  ^  4  ^(3  ^  _  i). 

5.  4  a;4  -  12  x^^  +  5  x22/2  +  12  xi/^  -^y^   and 

12  a;4  -  36  a:^^/ +  11  a;22/2+ 48  a:3/3  _  36  3^*. 

6.  7?2n(a:2  +  2/2)+  x?/(?n2  _|_  ^2)  ^nd  'mn{x^  +  3/^)+  xy{m^y  +  n2a:). 

7.  3  ax2  -  6  a2x  +  9  a3  _  3  a:2  +  6  ax  -  9  a2  and 

6  a2x2  +  24  aH  +  6  «*_  6  x2  -  24  ax  -  6  a\ 

8.  Show  that  the  proof  given  in  §  77  applies  to  expressions  contain- 
ing any  number  of  letters. 

9.  Explain  fully  the  method  of  finding  the  H.  C.  F.  of  more  than  two 
expressions. 

10.  Why  must  the  H.  C.  F.  of  any  number  of  expressions  be  a  factor 
of  the  H.  C.  F.  of  any  two  of  these  expressions  ?  Must  it  be  the  H.  C.  F. 
itself  of  any  two  of  the  given  expressions  ?     Explain. 

FindtheH.C.F.  of: 

11.  a4  +  4  a3  +  4  a\  a%  -  4  ah,  and  a%  +  5  a^^  4.  6  a%. 

12.  x8  -  6  x2  +  11  X  -  6,  x8  -  9  x2  +  26  X  -  24,  and  x^  -  8  x2  +  19  x  -  12. 

13.  a3  +  a2x-2x3,  a8+3a2a;+4ax2+2x3,  and   2  a8  +  3  02-^+2  ax2-2x8. 

14.  ax  +  }p-x  +  cH  -  acy  -  h'^cy  -  c^y,  a2  4.  2  a&  +  a62  ^  2  i^  +  00^+ 2  hc\ 
and  2  a2  +  2  alP-  +  c^h  +  2  ac^  +  6^  +  ah. 

79.*  Other  important  consequences  of  §  77.  Some  further  im- 
portant conclusions  may  be  easily  drawn  from  such  a  series  of 
divisions  as  that  described  in  §§  76  and  77 ;  thus,  if  iHf  and  N  are 
any  two  integers,  of  which  M  is  the  greater,  and  if  M  be  divided  by 
JV,  giving  a  quotient  Qi  and  a  remainder  i?i,  and  if  -^be  then  divided 
by  iJi,  giving  a  quotient  Q2  and  a  remainder  i?^?  ^iid  so  on,  —  sub- 
sequent quotients  and  remainders,  all  of  which  are,  of  course, 

*  This  article  may  be  omitted  on  a  lirst  reading. 


78-79]  HIGHEST  COMMON  FACTORS  121 

integers,  being  designated  by  Q^,  Q^,  Q^,  "-,  and  R^,  R^,  R^,  ••, 

respectively,  —  then  (§  47,  Ex.  11) 

•  M=Q,N'-\-R„  N=q,Ri+R2,  Ri=QsR2+R3,  ^2=9^+^4,  etc. 

From  this  series  of  equations  it  is  easy  to  express  the  several 
remainders  Ri,  Ro,  R^,  •••in  terms  of  M,  N,  and  the  quotients  Qi, 

Q,2j    ^3?    "•• 

Thus,  by  transposing,  the  first  equation  becomes  Ri=M—QiN., 
transposing  in  the  second  equation,  and  then  substituting  this  value 
of  Ri,  gives 

R,  =  N-q,R^=N-Q,{M-Q,N)  =  -q,M+(X+QiQ2)N) 

similarly,  from  the  third  equation, 

R^=R^-Q,R,={M-q,N)-q,\{i  +  q,q.:)N-  q,M] 
=  (1  +  Q2Q3)  M-  (Qi  +  4  +  q,q,q,)  n; 

and  so  on  for  the  later  remainders;  i.e.,  the  successive  re- 
mainders may  each  he  expressed  in  the  form  aM+  hN, 
wherein  a  and  b  are  integers  (one  positive  and  the  other  nega- 
tive), which  involve  the  successive  quotients,  hut  not  the 
given  numhers,  nor  the  remainders. 

Again,  if  M  and  N  are  prime  to  each  other,  then  (§  77)  the  last 
remainder  is  1,  and  therefore,  by  what  has  just  been  said,  two 
integers  a  and  h  can  be  found  such  that 

aM-\-hN=l. 

From  this  last  equation  it  is  easy  to  establish  the  following 
important  principle :  if  M  is  a  factor  of  NL,  hut  is  prime  to 
N,  then  it  is  a  factor  of  L. 

To  prove  this  it  is  only  necessary  to  multiply  the  above  equa- 
tion by  L ;  this  gives 

aML  +  hNL  =  L, 

wherein  the  first  member  is  manifestly  divisible  by  M  (M  being 
a  factor  of  NL  by  hypothesis) ;  therefore  the  second  member, 
viz.,  Z,  is  also  divisible  by  J/,  which  was  to  be  proved. 


122  ELEMENTARY  ALGEBRA  [Ch.  VIII 

EXERCISES 

The  following  direct  consequences  of  the  principle  just  now  established 
may  be  proved  by  the  student: 

1.  Ji  M  is  prime  to  N  and  also  to  L,  then  it  is  prime  to  the  product  NL. 

2.  ]f  ilf  is  prime  to  N,  L,  P,  •••,  then  it  is  prime  to  the  product  NLP  •••. 

3.  A  number  can  be  separated  into  but  one  set  of  prime  factors. 

4.  If  M  is  a  prime  to  N,  then  it  is  prime  to  any  integral  power  of  N. 

5.  Show  that,  with  slight  verbal  modifications,  the  principles  proved 
above  apply  also  to  integral  expressions  involving  one  or  more  letters. 

II.    LOWEST  COMMON   MULTIPLES 

80.  Multiples  of  algebraic  expressions.  A  multiple  of  an  alge- 
braic expression*  is  another  algebraic  expression  that  is  exactly 
divisible  by  the  given  one,  i.e.,  it  is  an  algebraic  expression  that 
contains  all  the  prime  factors  of  the  given  expression. 

A  common  multiple  of  two  or  more  algebraic  expressions  is  a 
multiple  of  each  of  these  expressions. 

E.g.,  12  a4a;3 (2/2—1)  is  a  common  multiple  of.3a'^x^{i/-{-l)  and  2a^z{y  —  l). 

The  lowest  common  multiple  —  usually  v^rritten  L.  C.  M.  —  of 
two  or  more  algebraic  expressions  is  that  algebraic  expression  of 
lowest  degree  which  is  exactly  divisible  by  each  of  the  given  ex- 
pressions; it  is  that  expression  which  contains  all  the  prime  factors 
of  each  of  the  given  expressions,  but  no  superfluous  factors. 

From  these  definitions,  it  is  easy  to  find  a  common  multiple  of  any  two  or  more 
algebraic  expressions  whose  prime  factors  are  known. 

E.g.,  a  common  multiple  of  a%'^x^  and  a^x^fj*  may  be  found  thus  : 

Since  a^  is  the  highest  power  of  a  that  is  found  in  either  of  these  expressions, 
therefore  any  common  multiple  of  the  given  expressions  must  contain  the  factor 
aS;  it  mar/,  of  course,  contain  a  still  higher  power  of  a.  Similarly,  a  common 
multiple  of  these  two  expressions  must  contain  &2,  x^,  and  y^  as  factors.  More- 
over, any  expression  which  contains  among  its  factors  a^,  b^,  x^,  and  y^,  is  exactly 
divisible  by  each  of  the  given  expressions,  and  is,  therefore,  a  common  multiple 
of  them. 

The  L.  C.  M.  of  these  expressions  is  that  one  of  their  common  multiples  which 
contains  no  factor  that  is  superfluous;  it  is  a%^x^y^. 

Similarly,  6  a^x^{x  —  2)^(x  — 1)3  is  a  common  multiple  of  a^x{x  —  2)^(x  —  1)  and 
x^ix  —  2){x  — 1)3,  but  it  is  not  their  L.  C.  M. ,  because  it  contains  the  factor  (x—2)^ 
when  only  (x— 2)2  is  needed,  and  it  contains  the  further  superfluous  factor  6;  the 
L.  C.  M.  of  these  given  expressions  is  a^x^{x  —  2)^(x  — 1)3. 

*  "Algebraic  expressioas  "  as  here  used  include  arithmeticalnumbers  also. 


79-80]  LOWEST  COMMON  MULTIPLES  123 

A  rule  for  writing  down  the  L.  C.  M.  of  two  or  more  monomials, 
or  of  any  two  or  more  entire  algebraic  expressions  ichose  prime 
factors  are  either  known,  or  can  easily  be  found,  may  be  formulated 
thus:  write  down  the  indicated  product  of  the  different 
prime  factors  that  enter  into  any  of  the  given  expressions, 
giving  to  each  of  these  factors  the  highest  exponent  which 
that  factor  has  in  any  of  the  given  expressions. 

EXERCISES 
Find  the  L.  C.  M.  and  H.  C.  F.  of 

1.  8  a%'^,  24  a'^b'^c^,  and  18  ahcK  5.  x"^  -  if-  and  x^  +  2  a:y  +  y\ 

2.  15  a%\  20  a%2c2,  and  30  ac^.  6.  21  a;^  and  7  x'^{x  +  1) . 

3.  16  aWc,  24  aMc,  and  36  a^W^.         7.  x^  -  1  and  x2  +  x. 

4.  V^a%r\  V^pYr,  and  54  a&y.  8.  ^x'^y-y  and  2  x'^  +  x. 

Find  the  L.  CM.  of: 
9.   a  +  i,  a  -  &,  a2  +  h"^,  and  a*  +  h\ 

10.  3  +  a,  9  -  a2,  3  -  a,  and  5  a  +  15. 

11.  x^  —  y%  x'^  +  xy  +  y^,  and  x'^  —  xy. 

12.  4  a  +  4  ^>,  6  a2  -  24  &2,  and  a2  -  3  a6  +  2  h\ 

13.  x^  +  y^,  x^y  —  ?/*,  and  x^  —  y^. 

14.  2/2  _  5  y  _^  6  and  ?/2  -  7  ?/  +  10. 

15.  x'^  ~  {a  -\-  b)x  +  ab  and  x'^  —  (a  —  b)x  —  ab. 

16.  Is  12  a%^(x'^  —  if)  a  common  multiple  of  2  a%{x  —  y)  and 
3  ab\x  -  y)  ?     Is  it  their  L.  CM.? 

17.  What  is  the  essential  requirement  in  order  that  one  expression 
may  be  a  common  multiple  of  two  or  more  others?  that  it  may  be 
their  L.  CM.? 

Find  the  L.  C  M.  of 

18.  3  x2  +  7  x  +  2  and  x"^  -  x  -  Q. 

19.  a2  +  4  a  +  4,  a^  -  4,  and  «4  _  16. 

20.  («  +  6)2  _  e2  and  (a  +  &  +  c)2. 

21.  a:'-"  -  ?/2n  and   (a:"  -  ?/")2. 

22.  a;3  +  6  a:2  +  5  a:  -  12  and  a,-3  -  8  a;2  +  19  a:  -  12. 
Suggestion.    Use  §  67  to  find  one  factor  of  each  of  these  expressions. 

23.  a;3  -  6  a;2  +  11  X  -  6  and  a;3  -  9  x2  +  26  a:  -  24. 

24.  o3  +  2  a2  -  4  a  -  8,  a8  _  ^2  _  8  a  +  12,  and  a^  +  4  rt2  -  3  a  -  18. 


124  ELEMENTARY  ALGEBRA  [Ch.  VIII 

81.  The  L.  C.  M.  of  two  entire  algebraic  expressions  found  by  means 
of  their  H.  C.  F.  The  use  of  the  H.  C.  F.  in  finding  the  L.  C.  M. 
may  be  better  understood  if  a  particular  example  be  first  worked 
out  before  the  general  discussion  is  given. 

Let  it  be  required  to  find  the  L.  C.  M.  of  Sx^  — x^  —  a?-\-x  —  2 
and  2  ar^  -  3  or  -  2  a;  +  3. 

By  §  76  it  is  found  that  the  H.  C.  F.  of  these  expressions  is 
a?  —  1]  they  may,  therefore,  be  written  thus : 

^x"^  -  a?  -  x"  +  x  -2  =  {x^  -l){^o?  -  X  +  2), 

and  2  x^  -^  x''  -  2  X  +  ^  =  (x"  -1){2  X  -  3), 

wherein  Za?  —  x  +  2  and  2 a?  —  3  have  no  common  factor.  Hence 
the  L.  C.  M.  of  the  given  expressions  is 

(a^  -  1)  (3  a^  -  .T  +  2)  (2  a;  -  3).  * 

Similarly,  in  general,  let  Ei  and  E2  be  any  two  entire  algebraic 
expressions,  and  let  their  H.  C.  F.  be  F)  then  they  may  be  written : 

and  E2  =  FQ2, 

wherein  Qi  and  Q2  have  no  common  factor,  since  F  is  the  H.  C.  F. 
of  El  and  E2.  Hence  the  L.  C.  M.  of  Ei  and  E2  is  the  product  of 
Ff  Qi,  and  Q2,  i.e.,  it  is  FQ1Q2. 

Moreover,  since  E^-  E2  =  FQi  •  FQ2  =  FiFQ^Q^,  therefore  the 
product  of  any  two  entire  algebraic  expressions  is  equal  to  the 
product  of  their  H.  C.  F  by  their  L.  C.  M. 

Hence :  to  find  the  L.  C.  M.  of  any  two  entire  algebraic  expres- 
sions, divide  the  product  of  the  given  expressions  hy  their  H.  C.  F. 

■r..    ,    ,      .    ^  ,,      .  EXERCISES 

Find  the  L.C.M.  of: 

1.  a:3  -  6  a:2  +  11  a:  -  6  and  x^  -  9  a:^  +  26  a;  -  24. 

2.  a;8-5  a:2  -  4  z  +  20  and  x^  ■\-2x'^-2ox-  50. 

3.  2  !/3  -  11 3/2  ^  18  ?/  -  14  and  2  ?/3  +  3  .y2  _  10  2/  +  14. 

4.  6  a^a;  -  5  a^x  -  18  ax  -  8  ar  and  6  a%  -  13  a'-b  -  6  ab  +  8  b. 

5.  4  x*  -  17  xY  +  4  ?/*  and  2  x*  -  xhj  -  S  x^y^  -5xy^-2  y\ 

6.  2  x4  -  9  a;3  +  18  x2  -  18  a:  +  9  and  3  a:*  -  11  a;^  +  17  a;2  -  12  a:  +  6. 

*  This  is  the  L.  C.  M.  because  it  contains  all  the  necessary  factors,  and  none 
that  are  superfluous. 


81-82]  LOWEST  COMMON  MULTIPLES  125 

82.   The  L.  C.  M.  of  three  or  more  expressions.     The  L.  C.  M.  of 

three  or  more  entire  algebraic  expressions,  whose  factors  are  not 
easily  determined,  may  be  found  by  first  finding  the  L.  C.  M.  of 
two  of  the  given  expressions  (§  81),  then  the  L.  C.  M.  of  that 
result  and  another  of  the  given  expressions,  and  so  on. 

T..    ,     ,      .    ^  ,,       .  EXERCISES 

Find  the  L.  C.  M.  of : 

1.  a;4  -  2  a;3  +  x2  -  1,  a:*  -  a;2  +  2  a:  -  1,  and  a:*  -  3  z^  +  1. 

2.  a:8  +  3  x^  -  G  ;r  -  8,  x3  -  2  a:2  -  X  +  2,  and  x^  +  x  -  Q. 

3.  a;2  -  4  a^,  a:3  +  2  ax^  +  4  cfix  +  8  a^,  and  x^-2ax'^^-^  a^x  -  8  a^. 

4.  If  A,  B,  and  C  stand  for  any  tliree  given  expressions,  and  if  31^  is 
the  L.  C.  M.  of  A  and  B,  while  M.j,  is  the  L.  C.  M.  of  M^  and  C,  prove  that 
M^  is  the  L.  C.  M.  of  A,  B,  and  C. 

Find  the  L.  C.  M.  of  : 

5.  a3  +  7  a2  +  14  rt  +  8,  a3  +  3  rt2  _  6  n  -  8,  and  a^  +  a^  -  10  a  +  8. 

6.  ^3  _  9  ^2  +  23  ^  -  15,  P  +  ^.-2  _  17  ^  +  15,  and  P  +  7  ^•2  +  7  ^  -  15. 


CHAPTER   IX 
ALGEBRAIC  FRACTIONS 

83.  Definitions.  An  algebraic  fraction  is  an  indicated  division 
in  which  the  divisor,  or  both  dividend  and  divisor,  are  algebraic 
expressions,  and  the  dividend  is  not  a  multiple  of  the  divisor. 

E.g.,  5^  ix  —  2y),  x^-^y,  and  Sax-i-  {a^  —  x^)  are  algebraic  fractions. 

Fractions  in  algebra  are  written  in  the  same  form  as  that  used 
in  arithmetic,  and  the  parts  are  called  by  the  same  names,  i.e., 
the  dividend  is  called  the  numerator,  the  divisor  is  called  the 
denominator,  the  numerator  and  denominator  taken  together  are 
called  the  terms  of  the  fraction,  and  the  numerator  is  usually 
written  above  the  denominator,  from  which  it  is  separated  by  a 
line. 

E.g.,  the  fractions  5-^  (x  — 2?/),  x^-^y,  and  3aa;-^  {a^  —  x^  are  usually  written 
^        ^^   and    f  "^   ■  respectively. 


x  —  2y     y  a^  —  x^ 

An  algebraic  fraction  is  called  a  proper  fraction  if  its  numerator 
is  of  lower  degree  than  its  denominator,  otherwise  it  is  called  an 
improper  fraction. 

E.g.,    ^  is  a  proper  fraction,  while _^'        is  an  improper  fraction. 

An  expression  which  consists  of  a  part  that  is  fractional  and 
a  part  that  is  integral  is  called  a  mixed  expression. 

E.g.,  m  +-,  a-\ —^-,  and  x-\-y —  are  mixed  expressions. 

p  a-\-  c  x  —  y 

Observe  the  difference  in  writing  a  mixed  number  in  arithmetic  and  a  mixed 

expression  in  algebra:  5|  means  5  +  |  in  arithmetic,  while  in  algebra  m-  means 

m  '  -,  and  not  m  +  -• 
P  P 

It  is  sometimes  desirable  to  write  an  integral  expression  in  the 
form  of  a  fraction;  this  is  done  by  using  1  as  the  denominator; 
e.g.,  a?  —  2x,  in  the  form  of  a  fraction,  is  ^"  ~     ^' 

126 


83-86]  ALGEBRAIC  FBACTIONS  127 

Attention  is  again  called  to  the  fact  that  algebraic  expressions  may  be  frac- 
tional in  form  and  yet,  for  certain  values  of  the  letters  involved,  represent 
integers,  and  vice  versa  [cf.  §  7,  (v)]. 

84.  Operations  with  algebraic  fractions.  As  in  arithmetic,  so  in 
algebra,  it  is  often  necessary  to  reduce  fractions  to  their  "  lowest 
terms"  and  to  a  "common  denominator,"  and  also  to  change 
mixed  expressions  to  improper  fractions,  and  vice  versa.  The 
operations  of  addition,  subtraction,  multiplication,  and  division 
must  also  often  be  performed  with  algebraic  fractions. 

Moreover,  since  algebraic  expressions  represent  numbers,  there- 
fore the  principles  which  were  demonstrated  in  §  54  apply  to 
algebraic  as  well  as  to  arithmetical  fractions,  and  all  of  the  above 
operations  are  therefore  essentially  the  same  in  algebra  as  in 
arithmetic ;  the  student  should  carefully  observe  this  similarity 
in  the  next  few  articles. 

85.  Converting  an   improper   fraction   into   a  mixed   expression. 

This  change  in  form  is  made  in  precisely  the  same  way  as  the 
corresponding  case  was  treated  in  arithmetic. 

E.g.,  just  as  V"  =  3^>  *-^-»  3  +  J,  so,  too,  since  a  fraction  is  an  indicated  divi- 

z^  +  z  +  1  x2  4-a;  +  i 

EXERCISES 

Keduce  each  of  the  following  improper  fractions  to  an  equivalent 
mixed  expression,  and  explain  your  procedure : 

^     a^-2ab  +  c  ^    a^  +  a^  +  1 


3a:  ' 

3    2  a:^  +  ax  -  3  a^  g 


4.  -"-I-IL^.  9. 

X  +  2 

5    x^-  a:8  -23;''-2a;-  1  ^q 

x'^-x-l  '  '  a:2-3x+l 

11.    Is    ^  ~  -^  + —  a  proper  or  an  improper  fraction ?    Why ? 
5  a^  —  8  a  +  3 


a 

3 

a:2  +  9  a;  +  2 

3a: 

2 

a:2  +  aa:  -  3  a2 

x-\-a 

t 

+  16 

a+1 

. 

8x3-  I0a;2-3a; 

+  5 

4  a2  -  3 

3  x6  +  2  X  -  5 

x8  +  2x+l 

7x6-1 

X3  +  X  +  1 

18  x4  -  X3  -  2  X2 

-7 

128  ELEMENTARY  ALGEBRA  [Ch.  IX 

Reduce  the  following  mixed  expressions  to  equivalent  improper  frac- 
tions, and  check  the  correctness  of  your  work  (cf.  Ex.  7,  §  39)  : 

12.  2x  +  ^^^1^^.  14.    :r  +  y  +  e  -  V^^^^- 

x^  +  >>  //  x^  —  y  —  z 

13.  Qy-x-\- — — 15.   3a-26  +  c-     4  +  *"^ 


4:y^  +  X  a  —  5  b  +  2  c 

86.  Reduction  of  fractions  to  lowest  terms.  In  §  54  (v),  it  was 
shown  that  any  factor  which  is  found  in  both  terms  of  a  fraction 
-may  be  rejected  (canceled)  without  changing  the  value  of  the 
fraction. 

3a^_3ax,        ,        x^-1      _{x-\-i)  (x-1)  ^x  +  1^ 
'^''  4.bxy^4:by'  x'^  —  2x-i-l      {x  —  l){x  —  l)      x  —  1 

In  algebra,  as  in  arithmetic,  a  fraction  is  said  to  be  in  its 
lowest  terms  when  the  numerator  and  denominator  have  no  com- 
mon factor;  hence,  a  fraction  may  always  he  reduced  to  an 
'^equivalent  fraction  in  its  loiuest  terms  by  dividing  both 
its  numerator  and  denojjiinator  by  their  H.  C.  F. 

fi  a'^x7/^ 
E.g.,  to  reduce  - — z'--  to  its  lowest  terms,  divide  both  numerator  and  denomi- 
9  ax^i/^ 

nator  by  3  axy^,  which  (§  73)  is  their  H.  C.  F. 

Instead  of  dividing  both  terms  of  a  fraction  by  their  H.  C.  F.,  and  thus  redu- 
cing the  fraction  to  its  lowest  terms  in  a  single  operation,  the  same  result  may,  of 
course,  be  accomplished  by  canceling  any  common  factor  as  soon  as  it  is  dis- 
covered, and  continuing  this  process  until  the  resulting  numerator  and  denomi- 
nator are  prime  to  each  other.  Recourse  to  the  H.  C.  F.  is  necessary  only  when 
no  common  factors  can  be  found  by  other  methods.  Observe  that  it  is  only 
equal  factors,  and  not  equal  parts,   that  may  be  canceled.       E.g.,      5^"t^- 

is  not  equal  to   |f  ;    nor  is  f^^^i^  equal  to   -^^^^.  '     "^ 

5&C  6s  — 5^2^  3  s  — 5  n^ 


EXERCISES 

Reduce  each  of  the  following  fractions  to  its  lowest  terms  : 
1.   ^^JZJI^.  4    a^  +  2ab  +  b^  ^  a^  +  b» 


&2  a8  +  63  a^  +  a2b'  +  b^ 

2    34  a^^c*  g  2  x^  +  3  a:  +  1  g     3  a^  _  2  a  -  1 

51  a^b^c  '  '     x'^+5x  +  4:  '                       '    l  +  a~a^-a^ 

^      ap.  -  yi  ^  a:«  +  y8^  ^      a*  -  ^2  -  20 


(a -6)2  y^-x^  a*-9a2  +  20 


85-87]  ALGEBRAIC  FRACTIONS  129 

10  a:^  +  2  xy  +  if  -  z^  ^5      Sfl^-f  OQa^  -  q  -  2 

z^  +  x^  +  7f-^2xij-^2xz  +  2yz         '   Sa^  +  17  a'-^  +  21  a  -  9* 

j_j_    a^-h^ ^g    a:5- 2x4 -2x3 -11x2-3;- 15 


12. 


a^  +  a^'ft  +  a^6"^  +  aW  +  b^  2  x5-7x4  +  4  x^- 15  x^-f  x-lO 

x3+3x2  +  4x+2  ^.y    a6(x2  -}-  y^)  .j.  a,-y(a-2  +  /j2-) 

x3-3x2-8x-10'  *   a&(x-'2-2/=2)+x^(a^-62)' 


j_3    x8  +  x2  -  22  X  -  40  j_Q    x8-6x2v  +  2x?/2  +  3?/3 


8-7x2+ 2x  + 40  x=^+0x2?/-2x?/'^-53/S 


j_^    1  -  2  X  -  5  x2  +  6  x3  ^g    a2  ^  52  _^  2  c2  +  2  a5  +  3  ac  +  3  ^»c 

*   l  +  5x+2x2-8x8'  '     a2  +  6^  +  c2+2a6  +  2«c  +  26c  ■ 

20.  May  the  factor  5  ax  be  canceled  from  the  first  two  terms  of  the 

numerator   and   denominator  of     5ax2- 10fl2^-  + 3  6(x -2  a)     ^-^^Yxout 

15  a3x4  -  30  a4x3  +  6  6(x  -  2  a) 
changing  the  value  of  this  fraction?    Why? 

21.  Is  the  value  of  a  fi-action  changed  by  canceling  equal /ac/ors  from 
both  numerator  and  denominator?  Is  it  changed  by  canceling  equal 
parts  or  equal  factors  of  parts  of  the  numerator  and  denominator? 

87.  Changing  fractions  to  equivalent  fractions  having  given  denomi- 
nators. Since  multiplying  both  terms  of  a  fraction  by  the  same 
number  does  not  change  its  value,  therefore  any  given  fraction 
may  be  reduced  to  an  equivalent  fraction  whose  denominator  is 
any  desired  multiple  of  the  given  denominator. 

E.g.,  to  reduce  j—^  to  an  equivalent  fraction  whose  denominator  shall  he 
12  cx^y,  multiply  both  terms  of  the  given  fraction  by  3  cy. 


EXERCISES 

1.  If  the  denominator  of  a  fraction  be  multiplied  by  any  given  ex- 
pression, what  must  be  done  to  the  numerator  in  order  to  preserve  the 
value  of  the  fraction  ? 

2.  How  find  the  expression  by  which  it  is  necessary  to  multiply  both 
terms  of  a  given  fraction  in  order  that  the  new  equivalent  fraction  shall 
have  a  given  denominator?    A  given  numerator? 

3  a  —  5 

3.  Reduce  - — — ^  to  an  equivalent  fraction  whose  denominator  is 

2  x(3  +  ax) 

18 X  —  2  a^x^.    Also  to  one  whose  numerator  is  12  ax  +  18  a?/  —  20 x  —  30 y. 


130  ELEMENTARY  ALGEBRA  [Ch.  IX 

Find  the  required  numerator  in  each  of  the  following  equations : 
4. 


3x-2a  ? 


5. 


x^-Sax+2a^     2x^- 7  ax^+7  a^x -2a^ 
4  ? 


(y  -  a)  (a  _  x)  (3  -  4  i/)      (a  -  y)  (a  -  a:)  (4  ?/  -  3)  (3  -  7  y) 


g    3m -8^  ?  ^    3x  ? 


2x-5      -2a:+5  1       7a;2-3a;  +  5 

88.   Reduction  of  fractions  to  common  denominators.     In  §  87  it 

is  shown  that  any  given  fraction  may  be  reduced  to  an  equiva- 
lent fraction  whose  denominator  is  any  desired  multiple  of  the 
given  fraction ;  if  then  any  common  multiple  of  the  denominators 
of  two  or  more  given  fractions  be  chosen  as  the  new  denominator, 
it  is  clear  that  these  fractions  may  be  reduced  to  equivalent  frac- 
tions having  this  denominator  in  common. 

E.g.,  since  12  a^x^  is  a  common  multiple  of  the  denominators  of  — ,  — ^,  and 

2x    3  .t2 

-— ,  therefore  these  fractions  may  be  reduced  to  the  equivalent  fractions  "  •^. 
o  a  12  a2a;2 

8  a^  10  amx^ 

^o  9  o '  ^^^  .,„  o  o  >  which  have  the  common  denominator  12 a^x^.  Similarly 
12  a^x^  12  a^x^  "^ 

for  any  given  fractions  whatever. 

In  practice  it  is  usually  desirable  to  keep  the  denominators  of 
fractions  as  small  as  possible,  and  therefore,  instead  of  choosing 
any  common  multiple,  as  above,  it  is  best  to  choose  the  L.  C.  M. 
of  the  given  denominators. 

2a  „„■,  5x 


E.g.,  the  L.  C.  M.  of  the  denominators  of — and 


(a;-l)(a;  +  l)  (x-\-l){x  +  3) 

is    (x  —  l){x-h  1)  (x  +  3) ,    and    these     fractious    are     respectively   equal    to 

ix-l)T+~l)l  +  B)  ^"^    i:c-l]T+l)l+S)  '  ^"''°^^'.'  *^^  ^^^^^  ^^"^^'""^ 
can  not  be  reduced  to  equivalent  fractions  having  a  lower  common  denominator. 

To  reduce  tvs^o  or  more  given  fractions  to  equivalent  fractions 
having  the  lowest  possible  common  denominator,  divide  the 
L.  C.  M.  of  the  ^iven  denominators  hy  the  denotninator  of 
one  of  the  given  fractions,  and  then  multiply  both  terms 
of  that  fraction  by  the  resulting  quotient;  do  the  same  with 
each  of  the  given  fra/itions. 


87-89]  ALGEBRAIC  FRACTIONS  131 


EXERCISES 

Reduce  the  following  fractions  to  equivalent  fractions  having   the 
lowest  possible  common  denominator : 

1.  3a+i  and  ^^±i.  5.  ^ and 


4                       6  (m-l)(m-2)           (2  -  wO(m  -  3) 

2.  ^-^^  and   ^  +  ^^.  6.        ^  +  y       and       ^  " -^     • 

16  6                 20  62  x'^  +  xy^y^           x^-xy  +  y^ 

3.  ±±^  and  ^Lll^.  7.  £^ll,    ^±i^,  and  ^'  +  ^^ 
a  —  b            a  +  h  x  +  y     x  —  y             x^  —  y'^ 

4.  -^^1-  and    ^  +  y.  8.    ^  " -^    and      ^' "  ^  +  ^    • 

X3  -  3/3                 3;3  +  2^3  a-S  _  2/3                 3.4.^  3.22/2  _^  y* 

9. ,    -,  and 


1  +  X     1  —  x^  x^  —  1 

10.        ^(--^'^      ,         ^(^^-^)     ,  and      ^« 


2  rt6  +  />'     a^  +  2ab+  b^  a^  -  b^ 

11.  6x 3a ^^^^      3a-  6x 


12.   7  a:, 


15  _  13  a:  +  2  0:2     a;^  -  8  a:  +  15  ^2  -  2  a:  -  15 

b  —  X       a  —  X      _    J  3 


:i.-2_62'  x2 -(a  + 6)a;  + a6 


13.  ^--'^      ,    ^_,  and      ^(^  +  ^)    . 

a:2+7a:  +  10     a;2  +  a:  -  2  a:2  +  4  a:  -  5 

14.  ,  ^  +  ^      ,      .    "-^ ,  and         « +  1 


a2  -  4  a  +  3'    a2  -  8  a  +  15'  a2  -  6  a  +  5 

^5^  5(u  -  3  .)^  8  and  ^^-^^). 

u  -  2  V        w2  —  5  My  +  6  y2  ^^  _  3  y 

89.  Addition  and  subtraction  of  fractions.  As  in  arithmetic,  so 
in  algebra,  the  sum  (or  difference)  of  two  given  fractions 
which  have  a  cormnon  denominator  is  a  fraction  whose 
numerator  is  the  sum  (or  difference)  of  the  given  numera- 
tors, and  whose  denominator  is  the  common  denominator 
of  the  §iven  fractions  [of.  §  54  (vii)]. 

E.g.       a^2  +  a;  +  5  a;2  +  3      ^a;2  +  a; +5-(a;2  +  3) 


Note  1.  The  minus  sign  before  the  second  fraction  means  that  all  of  that 
fraction  is  to  be  subtracted,  hence  the  necessity  for  the  parenthesis  in  the  numera- 
tor of  the  next  fraction. 


132  ELEMENTARY  ALGEBRA  [Ch.  IX 

Note  2.  Since  a  fraction  is  a  quotient,  therefore  its  sign  (i.e.,  the  sign  written 
before  the  dividing  line)  is  governed  by  the  laws  of  signs  in  division.  Thus,  if  Ei 
and  E2  are  any  algebraic  expressions  whatever,  then  —  — l  =  +  — ^  =  +  -^^. 

Hence  the  above  example  may  also  be  arranged  thus  : 

x'^  +  x  +  5  a;2  +  3      ^  x'^  +  x  +  b    ,     -a;2-3    ^       a;  +  2 

a;2-2a;  +  l     x^-2x  +  l     a;2_2a;  +  i     x'^-'lx  +  l     x^-2x  +  i 

If  the  given  fractions  have  7iot  a  common  denominator,  they 
must  be  reduced  to  equivalent  fractions  which  have  a  common 
denominator  (§  88)  before  they  can  be  added  or  subtracted. 

^         _1 3,      2     ^        xix  +  \)  3(a;-l)(a;  +  l)   I        2x(x-l) 

■•^■'    x-\      X      x  +  1      a:(x-l)Ca;  +  l)      a;(a;-l)(a;  +  l)      x{x-l){x  +  l) 

_x{x  +  l)—Z{x  —  \){x  +  \)  +  2x{x  —  l) 
x{x-l){x  +  l) 
^-x 
x{x-\){x  +  \)' 

3 3. 

and  this  result,  viz., ; — -,  is  called  the  "algebraic  sum"  of  the  given 

fractions.  x{x-l){x  +  l) 

EXERCISES 

,    Simplify  the  following  expressions : 

1  ^-^  ^  4. ±±A.  9       ^  +  "^  ^  +  ^ 


a;2  -  3  x  -  10      x2  +  2  X  -  35 

1   ^  re  +  3  J  a:  +  7,  ^^  1  3 


2  5  10 

3.  a-\-x  ^a-x^  j_3__ 

a  —  a;      a  -\-  x 

4.  1 +      ^  +  y— .  12. 

2  a;  -  3  y/     4  a;2  -  9  2/^ 

g    2  a;  —  3  g      2  x  -  a  je  -j^g 

a;  —  2  a         x  —  a 

6.  -3-  +  -1-.  14. 

a;  +  y     a;  —  y  a'-^  +  ax  +  a;'-^     a  +  a.- 

n    ^ 1^_  ,g       a  -\-h  +  c  a  —  h  -\-  c 


1  +  X  -  2  a;2      6  x2  -  a; 

-2 

^ 

1      +          1         . 

-  1      X  +  2  -  x2 

1            1 

(X 

-yy    {x  +  yy 

a  A-h                    a  ■ 

-b 

cfi 

-  2  a6  +  62     a2  +  2 

ab  +  b^ 

a2 

—  ax  +  x^     a  —  X 

x-e     x-2  a-2_(/;+c)2      (a-6)2-c'-2 

X  X  -  -    a  —  X  ,  a  +  X      0.2  _  2-2 


8.  ~t ± 16.   "  ~  ^  + 


1  —  x2      1  -f  x2  X  X  2  ax 


*  Compare  example  under  Note  2,  above. 


89] 


ALGEBRAIC  FRACTIONS 


133 


17     ^  ~  ^  1  ^  —  c      c  —  a 
ah  be  ac 

18.  2       3  ?/2  -x^  ^  XI/  +  f 
xy  xy^  x'^y^ 

19. h  3  a:. 

X  —  2  a         x  —  a 

Suggestion.    3  a:  =  — • 
1 

20.  ^       1         ^-y  3:2-a:.y 
X  -\-  y      x^  —  xy  -\-  y^      x^  -\-  y^ 

21.  UL^  +  lzi^  +  a:. 


1  —  a;      1  +  a: 


22. 
23. 
24. 
25. 
26. 
37. 


a;  —  a      a;^  —  a" 


a:(a: 

1 

-y) 

^Ri 

1 

a:?/ 

1 

1 

a:2- 

■7x- 

+  12 

a:2-  5 

a:-F  6 

2a;2 

1 

—  X 

-1^ 

1 
3-a;- 

2a:2 

1 

1 

2a 

27. 


28. 


29. 


30. 


31. 


32. 


33. 


34. 


35. 


a-  1 

a  (a  — 

1) 

1          2 

a      a  + 

l"^a  +  2 

1 

1 

x-1 

2(x  +  1) 

a 
a-  1 

2 
a  +  1      a 

a          1 

+  2      a 

1 

1 

a:  + 
2(x2  + 

:} 

1-x 

2(a:  +  1) 

1) 

7h- 

:.          ^' 

1-x 

2 

a:-3 

a.-3 

x  +  4. 

+  64 

b 

ai 

a62 

a  +  6      a  —  b      b^  —  a^ 

-1  1 


a  +  6      (a  +  />)2      (a  +  ft)  3 

a:^  +  ax^  _  x(x  —  a)  _    2  aa: 
aa;2  —  a^      a{x -\- a)      x^  —  a'^ 

x(a  -b)      b  -2i 
b^-x^  x  +  b 


36    ^^  -^  _  3  x(a  -  b)      b  -2a 


X  —  b 
1 


(a  -  ft)  (a  -  c)      (ft  -  c)  (ft  -  a)      (c  -  a)  (c  -  ft) 


Is  (rt  —  6)(a  — (^(^  —  c)  a  common  multiple  of  these  three  denomiuators?    Is 
(a  -  6)  (6  —  c)  (c  -  a)  ? 


38. 


1 


1   +   ^ 


X  —  1      a:  +  1   '  a-  —  2      a;  +  2  x^  -\-  y^      x^  —  y^       x^  —  y^ 


3  a;      ^  4  -  13  a: 


l  +  2a;      l-2a:      4a:2-l 
42. 


41.   ^-+     4«  1 


a 


1  +  2 


a;  +  a      x^  —  cfi      a  —  x     x^  +  a''^ 
3 


a;2-5a;  +  6      3a:-2-a:2      4a;-3-a:2 


43. 


44. 


a:  -  1  2 (a:  -  2) 


a:-3 


(x-2)(a;-3)      (3-a:)(x-l)       (a:  -  1)(2  -  x) 
a2  ,  ft2  c2 


(a-ft)(a-c)      (ft-a)(ft-0      (c-a)(c-ft) 


134                               ELEMENTARY  ALGEBBA  [Ca  IX 

45.  -^-^ +  ^« +  «* 


46. 


(a  -  c)  (a  -b)      (6  -  c)(b  -a)      (c  -  a)(c  -  b) 
1  2  ,  L 


x^  -  5  a:^/ +  6  2/2     x"^  -  ^  xy  -\- 3  y^     x'^-3xy+2 


^^    g^  +  2  g  +  1  _  2      g^  -  2  g  +  1        ^g    a:«  +  ar^  +  a:  +  1 3 ^ 

■g2-2g  +  l  g2  +  2g  +  l'  *        x^-ar  +  l  x-1 

90.  Reducing  mixed  expressions  to  improper  fractions.  Since  an 
entire  expression  may  be  written  in  the  fractional  form  with 
the  denominator  1,  therefore  reducing  mixed  expressions  to  im- 
proper fractions  is  merely  a  special  case  of  addition. 

'^''  x  —  1         1        x  —  1  z  —  1  x  —  1  x  —  1 

EXERCISES 

By  the  above  method  simplify  the  following  expressions : 
1.   .-l+^!_.  6.   3a-Sb      186^-5o« 


a;2  — 


g  +  2^> 

J.3      x*-\-x^-x  +  l 

1  +  a;  +  a.-2 

^       4  +  4y2+y3 

l-22/  +  2^2 

4  g2  +  9  62 
2a  +  36 

7        gx  +  6a:  +  g6 

2.   x  +  1-^^.  7.   a;-a;2-: 

x  —  1 

a:2-2a:  +  l  ^  ^ 

4.  l-y-yi-tuJl.  9.   2g-36 

1  -  2/4 

5.  g2  _  aft  +  ft2 ^ .  10.    1  -  ga:  -  ... 

rt  +  6  1  -  g6  +  a:2 

11.  Prove  that  any  mixed  expression  may  be  reduced  to  an  improper 
fraction  by  multiplying  the  integral  part  by  the  denominator  of  the  frac- 
tion, adding  or  subtracting  the  numerator  as  the  case  may  be,  and  placing 
this  result  over  the  denominator.     Also  compare  §  47,  Ex.  11. 

91.  The  product  of  two  or  more  fractions.  In  algebra,  as  in 
arithmetic,  the  product  of  two  or  more  fractions  is  a  frac- 
tion whose  numerator  is  tlie  product  of  the  numerators  of 
the  given  fractions,  and  whose  denominator  is  the  product 
of  their  denominators  [cf.  §  54  (ii)]. 

3x2     2a2?/        5rt62    _         ma^h'^T^y  Sa^b 


E.g., 


^by^     3x2     2x-Zy      12  bx^y^2 x - 3 y)      2yC2x-3y) 


ALGEBRAIC  FRACTIONS  135 


i'in 

d  the  product  of : 

EXERCISES 

1. 

«^^     and   ^'< 
b^dh-^             ab-2 

3.  f ":    and    *"'"^ 
6'»+2               a-" 

2. 

Sxy   ^^^   16  ,V 
8yz             9xY 

4.       "and       *    . 
a  +  6            a  -  6 

5. 

«'-«^   and  ^'+^-^. 
x^  —  xy            a^  +  ab 

6. 

! and   — 

'  -  xy  +  y'^ 

X*  +  x-y'^  +  y^ 

(a  -  by 

7.  Simplify  (x  -\-2y  —  )      ^   ,  making  use  of  the  distributive  law. 

\  yl  a  +  a; 

8.  Simplify  (x  +  2y  —  ] — ^  by  first  reducing  the  multiplicand 

\  y/a  +  X 

to  an  improper  fraction  (cf.  Ex.  7). 

9.  Simplify    f  ?/  +  3 ^]  f  2  ?/  +  3  -  — -^ — ]  by  the  method  of 

\  y  —  0/  \  2  y  —  '6/ 

Ex.  7,  and  also  by  that  of  Ex.  8,  and  compare  results. 

10.  Give  a  convenient  rule  for  multiplication  when  one  or  more  fac- 
tors are  mixed  expressions. 

p     r      pr 

11.  Prove  that  —.-  =  —;-,  and  show  that   the   proof   is  still  valid 

when  some  or  all  of  the  letters  represent  algebraic  expressions  (cf.  §  54 
and  §  84). 

12.  How  may  an   integral  expression  be  multiplied  by  a   fraction 
(cf.  §  54)  ?    Is  n  .  -  equal  to  —  ?    Is  it  equal  to  -^— ? 

13.  How  does  the  identity  I^Y  =^  follow  from  §  54  (ii)? 

14.  Prove  that  ^S.=  P^,  and  thus  prove  that  El  .  iL  =  P/: .  A  =  2L. 

q       qn  qs     rw       q^    /w      qio 

15.  Based  upon  Ex.  14,  give  a  convenient  rule  for  multiplying  two  or 

more  fractions  together  by  cancellation. 

Find  the  product  of : 


16. 

^«-^^'and     ^'+-y^y' 

19. 

^'^y'  and     «*-«^'^^  +  ^\ 

x^  —  y^             a^  —  "2  ax  +  x^ 

a6  +  66             x^+2  a;23/2  +  / 

17. 

i^-^y-^   and  «  +  ^+l. 
(a  +  6)2_i             a -6-1 

20. 

^'  +  ^^   and    f     ^  -         y     ' 
a:2+y2              \x-y      x^y 

18. 

«'-!   and    («^+l)(^  +  l)^ 

21. 

a+     «*      and    b         "^   • 

a-\-b 


136  ELEMENTARY  ALGEBRA  [Ch.  IX 

x-^-9x  +  20  a;2  -  6  X  +  9 

(ga-DCaS+l)    ^^^  (a-DHa  +  iy       , 

x^  +  x-^?/=2  +  y^  (a2  -  1)  (a4  _  2  a2  +  1) 

24    «iz^8^  and  «  +  -^        - 

'     -^      4  62  a2  +  2  a6  +  4  62 


25.  5!^ll?,  ££:L  +  1^,  and  -liii- 
2  xy  XT/  +  2/2  x^  —  xy 


xy  +  2/2  aj2  _  2:^ 

26    "^  +  6'^  -  c2  +  2  aft    ^^^   ^2  _  ^,2  ^  ^2  _  o  ^^ 
•   (,2  _  j-2  _  c2  _  2  he  c^-a^+}p--  2  6c* 

27.  ^^JLi  +  ^LZL^   and  ^L±i  _  ^^IL:^. 
a  —  6      a +  6  a  —  6      a  +  6 

28.  «  -A-l--?    and    1 ^-^—' 

he      ac      ah      a  a  +  6  +  c 

92.  Division  of  fractions.  In  algebra,  as  in  arithmetic,  to  divide 
hy  any  fraction  gives  the  same  result  as  to  multiply  hy  the 
reciprocal  of  that  fraction  [cf.  §  54  (vi)]. 


■^*'    b^y^  '  bijlf^'  c»s   bey 


Note.  If  the  divisor  is  an  integral  expression,  it  should  be  first  written  in  a 
fractional  form,  and  if  it  is  a  mixed  expression,  it  should  be  first  reduced  to  an 
improper  fraction,  before  proceeding  as  above. 

EXERCISES 

1.   Prove  that   -2  -  ^  =  ^  .  !  (cf.  §  54  and  §  84). 
q      s      q     7- 

Perform  the  following  indicated  operations,  and  simplify  the  results  : 
2      Qx^y    .    2x^  -     /        3x2W  ^2       A  .   a 

14a364  ■  2a262"  '    V        a    JVSx^        )  '  x'' 

(q  _  />)2  _  p      q  _  ;,  +  3 
(a  +  6)2  -  9  ■  a  +  6  +  '6 

x2-  1  .y2-  12  X  +  8.5 

x2-  ;}x  -  10  ■     x2  +  3x  +2  ' 

<72  -I-  .r-2  _  1  4.  9  ax  .  g  +  1  +  X 
x2  +  y-  —  d  -{-  2  xy      x  +  3  +  y 

3:3  _   n  y2  4.  30  y  x^  +  216x 

x2  -  49  '  x2  -  X  -  42' 


3.    "    -  ^-^^  ^^  -r  ^-^,  Q 


14  a364  ■ 

2a262 

a2  _  121 

.  a  +  11 

a2-4 

a  +  2 

X3  -  «8 

(x-«)2 

x3  +  a3 

x2-a2 

14x2- 

7x        2x- 

-  1 

12  x3  +  24  x2  ■  x2  + 

2x 

a*- 

64         .  («  - 

-6)« 

a4  +  a26' 

2  +  ¥       «« 

-6« 

5.  "^-^ Lj^  ^  ^uf: i..  10. 

12  x3  +  24  x2      x2  +  2  X 

6.  «^-^^        ^  (^^  -  ^>)"  11. 


91-93]  ALGEBRAIC  FRACTIONS  137 

i  m^n  —  5  n"^  m^  —  mn  +  n^ 


IS- 
14. 


2  x^  +  13  a:  -f  15  .  2  a:^  +  11  r  +  5 
4  x2  -  9  4  a;-^  -  1 


a*  -  84  a"'^  +  25     8  a-^  +  8  a  +  5 


15    4  g-^  +  />2  _  ^2  ^  4  ,,ft      2  g  +  ^  +  c  ^ 

4  rt--^  -  //2  _  c-^  -  2  6c  ■  2  a  -  6  -  c'  ■ 


16. 


17. 


a^  -\-  ah  -]-  ac  +  he  ^  a^  —  ax  +  ay  —  xy      a^  —  a(y  —  h^  —  hi) 
ax  —  ay  —  x'^  -^  xy     d^  +  ac  +  ax  -\-  ex     x"^  —  x(y  —  a)  —  ay 

a:4  -  .3  a:3  -  28  x^  +  75  x  -  50     .  3:3  -  12  a:^  +  45  -g  _  50 
a;4  _  5  a;3  _  21  a;2  +  125  a:  -  100  "  a:^  _  10  a:^  +  29  a;  -  20' 


18.    P^-^*  ^  P^  +  P^  +     P^     .  p--^P9  +  g^ 

iP  -  Q)^       P  -  Q       P^  +  (f     P^  +  2/'^  +  <?2 

93.  Complex  fractions.  In  algebra,  as  in  arithmetic,  a  fractioii 
whose  numerator  or  denominator,  or  both,  are  themselves  frac- 
tional expressions,  is  called  a  complex  fraction. 

\-a             z-l 
E.g.,  .,   I       and are  complex  fractions. 

^^^  x  +  2-\-- 

X 

A  complex  fraction,  like  a  simple  one,  is  primarily  an  indicated 
quotient,  but  it  usually  also  involves  some  of  the  other  funda- 
mental operations  alread}''  studied;  performing  these  operations 
is  spoken  of  as  simplifying  the  fraction. 

E.g.,  the  above  complex  fractions  are  simplified  as  follows: 


\      and 


l-a 
a 

1  +  a 

={-l-«)- 

-(!+«)  =  - 

-«2 

a 

1  + 
1 

a     1- 
a 

cr2        1         1 
1  +  a 

—  a 
a 

.-1 

X 

x2_i 

a;2  +  2  x  + 1 

'  =  ?! 

-1^ 

X 

X 

x-\ 

a;  +  2  +  i 

x^-\-2z 

+  1     K  +  l 

Note.    Multiply  both  numerator  and  denominator  of  this  last  fraction  by  x, 
and  reduce  to  lowest  terras.    How  does  this  method  compare  with  that  used  above  ? 

*  To  avoid  ambiguity,  the  principal  dividing  line  in  a  complex  fraction  is  best 
made  somewhat  heavier  than  the  others. 


138 


ELEMENTARY  ALGEBRA 


[Ch.  IX 


EXERCISES 

Simplify  each  of  the  following  expressions  : 


[Va^/       a6      (a-6)2j    [    [  \a      bl       ab 

2  /^  ^    I    M  -  h  •  f  ^-^       Ml- 

I      ^a  +  6^  (a  +  />)2||  (a +  6)2  I 


1  - 


rrr  +  n' 


n      m 

1  + 


(a  +  by 


m2-  n2 
m^  —  n^ 


1  + 


^) 


6. 


X  -  y 

X  +  y 
-  2  - 


a  -  3  i)  J  I 


1  + 


26-a 


^  + 


X  + 

1 


xy 


x^  + 


x^  —  y' 


X  +  y  } 


10. 


a  —  6      a  +  b 


11. 


X  —  ^  - 

a: 

-4 

^      4  - 

1     ' 

a: 

-4 

fl?;2 

4  rr/>(a  +  ft) 

3  +  &3 

a2  - 

-  a6  +  62 

a2  +  ft2 

a2 

-62 

a2-ft2 

a2 

+  62 

a  +  6 

a 

-6 

12. 


m  —  n      m-^  +  n' 
m  +  n      m2  —  71^ 


13 


6      a  +  6 

^8    _    yZ 


a  —  b     a  +  6 


7n2n  +  n* 


m  —  n      (m  —  ny 


x^  —  xy  -\-  ^2     X 


x^  +  y^       .  (l  ^     y    \ 
'^  +  xy  +  y^     \        X  -  yl 


98]  ALGEBRAIC  FRACTIONS  139 

9  a:2  -  64 


14.   1  + 


17. 


I  +  X  + 


O  -r^  '  „  1  1 


X-   1 


1-a;  1 ^ 

4  +  a: 
1 


18. 


,^    3x-2     Sx  +  2 

15.  _ 1  ^ 


^^  1  -  o: 

^  +  Ui  19 £±iL 


1^  «   ^    <^  ,     , 

16.  J a;  +  3/  + 


a   ,   b   ,  c  ,1 

H h-  X  -  y  + 


b      c      a  X  -\-  y 

20.    Recalling  the  meaning  of  a  negative  exponent  (§  44),  show  that 
,4.1 


cr- 

^2      -       ^,2        -^2^3* 

21.  As  in  Ex.  20,  show  that  _'?!r^lV  ^  a^ffi,     gj^o^  ^jgo  l^h^t  ^ 
^mjy-n  m^s-^w^      m'^wH^  by 

~  b^s-"' 

22.  Prove  that  any  factor  whatever  of  the  numerator  of  a  fraction 
may  be  transferred  to  the  denominator  by  merely  reversing  the  sign  of 
the  exponent  of  that  factor.  Also  show  how  a  factor  may  be  transferred 
from  denominator  to  numerator. 

23.  Is   ^^  +  ^~^^^  equal  to  9^+^'?    Why?    Observe  carefully  that 

4  X  4  xb^ 

a  factor,  but  not  a  part,  may  be  transferred  as  in  Ex.  22. 

Clear  the  following  expressions  of  negative  exponents,  and  simplify 
them  as  far  as  possible;   in  any  case  of  doubt  employ  the  definition 

of  §  44,  viz.,  a~*  =  —  • 
a* 
24      3  m-^n^  25  ^'^^  26    3a  +  2ftc-8 

'  2  (a  +  a:)*  '   3  •  2-^b^y-^'  '   5  x-^  -  H  y 


REVIEW  QUESTIONS-CHAPTERS  VI-IX 

1.  Define  and  illustrate :   even  numbers;  odd  numbers;  prime  num- 
bers; composite  numbers;  finite  numbers;  and  infinite  numbers. 

2.  What  is  the  value  of  g?    Of§?    Explain. 

3.  Show  that  the  absurdity  in  Ex.  17,  §  55,  arises  from  dividing  zero 
by  zero. 


140  ELEMENTARY  ALGEBRA  [Ch.  IX 

4.  By  applying  the  distributive  law  show  that  —  (a  +  x  —  5)  =  —  a 
-  a:  +  5. 

5.  State   the   binomial   theorem.     Apply  this   theorem   to  expand 
(2  a  -  3  x^y. 

6.  If  Ax"^  +  Bx"^-^  +  •••  +  Hx  -\-  K  is  divided  by  x  —  a,  prove  that 
the  remainder  is  Aa"^  -\-  Ba"-'^  +  ■-  -\-  Ha  -{■  K. 

7.  By  means  of  Ex.  6,  and  without  actually  performing  the  division, 
show  that  a:  —  1  and  x  +  2  are  factors  of  x*  +  2  x^  +  7  x  -  10. 

8.  As  in  Ex.  7,  show  that  x"  -  ?/",  wherein  n  is  any  positive  integer 
is  exactly  divisible  hy  x  —  y. 

9.  By  means  of  factoring,  find  the  roots  of  x^  —  7  x  +  12  =  0,  and 
explain. 

10.  Form  the  equation  whose  roots  are  3  and  —  7,  and  explain. 

11.  What  is  meant  by  the  L.  C.  M.  of  two  or  more  expressions?    How 
may  it,  in  general,  be  found? 

12.  How  may  the  L.  C.  M.  of  three  or  more  given  expressions  be 
found? 

1+      ^ 


13.  Simplify  l  +  x^C^-D-^. 

^     ^        ,       1  x^  +  x  +  l 

X  + ^     ^ 

a:  +  1 

14.  Is  2rtx5^  g       J  ^^  a^ — J    Explain. 

3  63        ^  3-2-163^-2  ^ 


CHAPTER   X 

SIMPLE  EQUATIONS 

I.    INTEGRAL  EQUATIONS 

94.  Introductory  remarks  and  definitions.  Some  preliminary 
work  in  simple  equations  has  already  been  given  in  Chapter  III ; 
the  text  of  that  chapter  should  now  be  rapidly  reread.  In  the 
present  chapter  it  is  proposed  to  treat  this  subject  in  a  somewhat 
more  careful  and  rigorous  manner. 

Every  algebraic  problem  involves  one  or  more  numbers  whose 
values  are  at  first  unknown,  and  which  are  to  be  found  from 
given  relations  which  they  bear  to  other  numbers  whose  values 
are  known ;  to  distinguish  between  these  two  kinds  of  numbers 
the  first  are  called  unknown  numbers,  and  are  usually  represented 
by  some  of  the  later  letters  of  the  alphabet,  as  x,  y,  and  z  (cf. 
§  26),  while  the  others  are  called  known  numbers,  and  are  repre- 
sented either  by  the  Arabic  characters,  1,  2,  3,  •••,  or  by  some  of 
the  early  letters  of  the  alphabet,  as  a,  b,  and  c. 

If  any  of  the  known  numbers  in  an  equation  are  represented 
by  letters,  then  it  is  called  a  literal  equation,  otherwise  it  is  called 
a  numerical  equation.  If  its  members  are  integral  expressions  so 
far  as  the  unknown  numbers  are  concerned  (§  41),  then  it  is 
called  an  integral  equation ;  known  numbers  may  appear  as 
divisors  and  the  equation  still  be  integral. 

E.g.,  3x2  +  5x1/ -10 2/2  =  8,  4-^  =  7x,  and  5(x2  +  2/2)=^  are  integral 
equations;  the  first  two  are  numerical,  while  the  third  is  literal. 

By  the  degree  of  an  integral  algebraic  equation  is  meant  the 
highest  number  of  unknown  factors  which  it  contains  in  any  one 
term.  If  all  of  its  terms  are  of  the  same  degree,  the  equation  is 
homogeneous. 

141 


142  ELEMENTARY  ALGEBRA  [Ch.  X 

E.g.,  3x4-7  =  13  and  2  +  4?/  — 5x  =  0  are  numerical  equations  of  the  first 
degree,  while  z^  +  10x  =  ^x  —  ^,  4 xi/2  =  3 ax^  —  7 ?/3,  and  azy^  —  x  =  3y  are  of 
the  third  degree ;  of  these  last  three  equations  the  first  is  numerical,  the  second 
and  third  are  literal,  and  the  second  is  homogeneous. 

Special  niimes  are  often  given  to  equatioils  of  the  lower  degrees;  thus  an 
equation  of  the  first  degre6  is  known  as  a  simple  equation  and  also  as  a  linear 
equation ;  *  one  of  the  second  degree  is  also  called  a  quadratic  equation ;  one 
of  the  third  degree,  a  cubic  equation;  etc. 

EXERCISES 

1.  What  is  meant  by  a  root  (or  solution)  of  an  equation?  Is  2  a 
root  of  a:2  -  7  a:  +  10  =  0  ?  What  then  are  the  factors  of  x'^  -  7  x  +  10 
(of.  §  67)?    What  other  root  has  this  equation? 

2.  Verify  that  x  =  4:  and  y  =  3  constitute  a  solution  of  the  equation 
7  X  +  2  y  =  M.  If  a:  =  2  in  this  equation,  what  must  be  the  corresponding 
value  of  y?     If  a:  =  a,  what  is  y  ?     If  y  =  6,  what  is  x?     Find  four  other 

.solutions  of  this  equation. 

3.  How  many  solutions  has  the  equation  in  Ex.  1  ?  How  many 
solutions  has  the  equation  in  Ex.  2? 

4.  Is  the  equation  in  Ex.  1  homogeneous?  integral?  literal?  numeri- 
cal? simple  ?     Define  each  of  these  kinds  of  equations. 

5.  Show  that  x^  +  lOx'^y  +  8^^  =  dxy^  is  a  homogeneous  equation. 
What  is  its  degree  ?  Can  a  homogeneous  equation  have  a  term  free 
from  the  unknown  number  ? 

6.  Is  3x2  —  5?/2  =:2a2  homogeneous?  Why?  Write  a  homogeneous 
linear  equation  in  two  unknown  numbers ;  also  an  integral,  literal,  quad- 
ratic, non-homogeneous  equation  in  two  unknown  numbers. 

Solve  the  following  equations,  using  the  methods  of  Chapter  III,  and 
also  §  72 : 

7.  ^-'^^^Ji^  +  5  =  0.  10.   2ax  =  2c-3bx. 

8.  x-Sx  +  4:-(sx+2-'f\  =  0.       11.   -^--^  =  c. 

\  4:1  2a     46 

9.  ^^-=i-^^;^+2=0.  12.    {a-x){a-h)-a{h-x)  =  (). 

13.   x^-x  =  Q.  14.   x2  +  (a  -  6)r  =  «&.  15.   x^  ^2  x"- =  x  ^2. 

16.    Find  three  solutions  of  .5  a:  —  3  y  =  7. 

*  The  appropriateness  of  this  name  will  be  seen  in  §  115. 


94-95]  SIMPLE  EQUATIONS  143 

95.  Equivalent  equations.  Two  equations  are  said  to  be  equiva- 
lent if  every  root'  of  eitlier  is  also  a  root  of  the  other. 

The  methods  thus  far  employed  for  solving  equations  (in  Chap- 
ter III,  and  elsewhere)  consist  in  clearing  equations  of  fractions, 
transposing  and  collecting  terms,  etc.,  i.e.,  these  methods  consist 
in  deducing  from  any  given  equation  a  succession  of  iiew  equations 
whose  roots  are  more  and  more  easily  found,  and  then  finding  the 
root  of  the  simplest  of  these  new  equations,  —  compare  Exs.  1  and 
2,  §  24. 

That  the  root  of  this  final  simplest  equation  happens  also  to  be 
a  root  of  the  given  equation  depends  upon  the  following  prin- 
ciples :  , 

(1)  Adding*  the  saine  numher  to  each  member  of  any 
given  equation,  forins  a  new  equation  which  is  equiva- 
lent  to   tJie  first  (cf.  §  24,  Ax.  1). 

(2)  Multiplying*  each  member  of  an  equation  by  the 
same  number  or  algebraic  expression,  which  does  not 
involve  the  unhnown  number,  and  which  has  a  finite 
value  different  from  zero,  forms  a  new  equation  which 
is  equivalent  to  tlze  first  (cf.  §  24,  Ax.  2). 

To  prove  Principle  (1)  let  the  member^fVny  fei\^eu  equation  be  represented 
by  El  and  E^  respectively,  i.e.,  let  the  equ^iott  be   ^ 

Ei^E^    \    \  (1) 

This  does  not  mean  that  Ei  and  E.2  represent  the  same  number  for  every  value 
that  may  be  substituted  for  the  unknown  nurnbet^  but  that  they  represent  the 
same  number  only  when  a  root  of  the  equation  if  substituted  for  the  unknown 
number. 

But  manifestly,  if  N  represents  any  nuntfipn whatever,  then 

Ei  +  N=B^^N    I    ^  (2) 

whenever  Ei  =  E^;  i.e.,  every  root  of  Eq.  (1)  is  also'a  root  of  Eq.  (2). 

By  precisely  the  same  reasoning,  every  rootr^  Equation  (2)  is  also  a  root  of 

(El  -{-N)  +  (-N)  =  (E2  +%  +  (-  -ZVT),  (3) 

i.e.,  of  Ei  =  E<i. 

Hence,  every  root  of  Equation  (1)  is  a  root  of  Equation  (2),  and  vice  versa; 
therefore  these  equations  are  equivalent. 

*  Since  adding  a  negative  number  is  the  same  as  subtracting  a  positive  number 
of  the  same  absolute  value,  and  since  dividing  by  any  number  is  the  same  as 
multiplying  by  its  reciprocal,  therefore  subtraction  and  division  are  included  in 
these  statements. 


144  ELEMENTARY  ALGEBRA  [Ch.  X 

To  prove  Principle  (2),  it  is  simpler  first  to  write  Equation  (1)  in  the  form 

^1-^2  =  0,  •  (4) 

which,  by  Principle  (1),  is  equivalent  to  Equation  (1). 

If  now  N  represents  any  finite  number  that  does  not  contain  the  unknown 
number,  and  is  not  zero,  then  manifestly 

N{Ei-E2)=0  (5) 

for  every  value  of  the  unknown  number  which  makes  Ei  =  E^,  and  for  no  others, 
i.e.,  every  root  of  Equation  (1)  is  also  a  root  of  Equation  (5),  and  vice  versa  ;  i.e., 
Equation  (1)  and  Equation  (5)  are  equivalent. 

Note  1.  That  the  multiplier  in  Principle  (2)  above  must  not  contain  the  un- 
known number,  and  that  it  must  not  be  zero,  becomes  evident  on  examining  any 
given  equation,  e.g.,  3a;—  4  =  2.  On  multiplying  each  member  of  this  equation 
by  cc  — 3,  and  simplifying,  it  becomes  3x2  — 15a;  +  18  =  0;  but  since  3  is  a  root 
of  this  equation,  and  not  a  root  of  3  a;  — 4  =  2,  therefore  the  two  equations  are 
not  equivalent. 

So,  too,  if  each  member  of  the  given  equation  be  multiplied  by  zero  it  becomes 
(3  a;  —  4)  •  0  =  2  •  0,  of  which  any  finite  number  whatever  is  a  root,  and  hence  the 
new  equation  is  not  equivalent  to  the  given  one. 

Note  2.  The  language  in  this  discussion  applies  to  equations  containing  only 
one  unknown  number,  though  it  is  evident  that  the  same  argument  is  applicable 
however  many  unknown  numbers  may  be  involved. 

EXERCISES 

1.  Apply  Principles   (1)  and  (2)   of   §  95  in  solving  the  equation 

^^-^ H  ^  ~  '  +  3  X  =  0 ;  and  show  in  detail  that  each  derived  equation 

2  o 

is  equivalent  to  the  one  preceding,  and  thus  to  the  given  equation. 

2.  Show  that  the  equation  6  a:  —  30  =  — ^^-^ — -  +  36  is  equivalent  to 

3  X  —  '^ 
X  —  5  = +  6 ;  and  that  each  of  these  is  equivalent  to  7  a;  —  35  = 

3  X  —  2  +  42,  and  therefore  to  4  a:  =  75,  i.e.,  to  a:  =  18|. 

3.  Provided  that  no  error  has  been  made  in  the  transformations  in 
Ex.  2,  do  we  really  know,  without  verifying,  that  18|  is  a  root  of  the 
given  equation  ?  *     Why  ? 

4.  Show  that  Principle  (1)  above  includes  the  principle  of  transposi- 
tion (§  25)  ;  and  that  Principle  (2)  is  far  more  useful  in  solving  equations 
than  Axiom  2,  §  24  (cf.  Note  1,  §  95). 

*  Though  it  is  no  longer  necessary  to  verify  that  the  root  of  the  last  of  such  a 
set  of  equations  as  those  in  Ex.  2  is  also  a  root  of  the  given  equation — because 
of  the  principles  of  §  95 — yet  verifying  serves  as  a  check  upon  the  correctness  of 
the  actual  work,  and  is  still  recommended. 


95-97]  SIMPLE  EQUATIONS  145 

Apply  Principles  (1)  and  (2)  of  §  95  in  solving  the  following  equations ; 
and  in  particular  point  out  the  equivalence  of  the  several  equations 
involved  in  each  exercise,  and  the  reason  for  this  equivalence  : 

f.    X      2  a;  —  ;3,3x  —  15      o  ik 

5.- ^-  +  ^ 2x^15. 

6.  3  a;  -  3(2  a;  +  15)  +  2(a:  -  2)  -  14  =  0. 

7.  (3  a;  -  5)  (a;  -  2)  -  4  a;2  +  14  a;  -  12  =  0. 

4a;-5      7  a:  -  15      4(3  a: -2)         ,«7a:  +  l      ,,o         4a;  +  7 


2       ~       3 


7i  -  a:     a:/6 


I(.-0- 


14.   1.75:.+3+:5£=:26£,z2:375. 
.25  1.125 

15.  ^i;il+ 1-^1^  =  0. 


96.  Literal  equations.  The  same  method  that  has  been  followed 
in  the  solution  of  numerical  equations,  and  the  same  principles  as 
are  there  involved,  apply  also  to  literal  equations. 

E.g.,  given  the  equation  ax-{-h  =  cx+d;  to  find  x.  This  equation  is  equiva- 
lent to  ax  —  cx  =  d—h,  [§  95  (1) 
i.e.,  to                                    x{a  —  c)  =  d—  b, 

and  hence  to  x  =  ^  ~    ,  rs  95  (2) 

a  —  C  La  \    J 

which  is  the  required  root. 

Show  that  the  root  just  found  will  serve  as  a  formula  for  solving  any  equation 
of  this  kind  [cf.  §  9  (ii)]. 

97.  A  simple  equation  in  one  unknown  number  has  one  and  but 
one  root.  By  transposing  and  uniting  terms,  etc.,  every  equation 
of  the  first  degree,  which  is  not  an  identity,  and  which  contains 
only  one  unknown  number,  is  easily  reduced  to  an  equivalent 
equation  of  the  form  ax=b  (§  95) ;  but  this  last  equation  has, 
manifestly,  one  and  but  one  root,  viz.,  b  h-  a,  hence  the  given  equa- 
tion has  one  and  but  one  root,  which  was  to  be  proved. 


146  ELEMENTARY  ALGEBRA  [Ch.  X 

EXERCISES 

1.  What  is  a  literal  equation  ?  A  numerical  equation  ?  To  which 
class  does  2x  —  V6  +  ax  =  14:x  belong  ? 

2.  Find  the  roots  of  a;^  —  5  a:  +  6=  0.  How  many  roots  has  this 
equation?  By  factoring  its  first  member  pi'oce  that  this  equation  has 
the  two  roots  3  and  2,  and  that  it  has  no  other  root  whatever. 

3.  How  many  loots  has  3a;  —  l  =  a:  +  3?  How  do  you  know  that 
it  really  has  one  root?     Prove  that  it  has  only  one. 

4.  By  the  formula  of  §  96,  solve  the  equation  in  Ex.  3,  and 
explain  fully. 

Solve  the  following  literal  equations;  show  in  detail  that  the  steps 
you  employ  always  yield  equivalent  equations ;  verify  the  correctness  of 
your  solution  in  Exs.  5-10  by  actual  substitution  of  the  roots.  Also 
solve  Exs.  5-8  by  using  the  formula  of  §  96  : 

X  —  2  ab      -,       a:— 3c 


5.   bx-(a  +  b)x  +20-cx  =  d.  8. 


c  ab 


6. 

C8- 

-  X  +  n^x 

= 

7. 

X 

b 

x  +  2b 
a 

b 

n-c-x.  9,   7  x+5(l-—]  =  a(x-a). 

_  3  10    ^  ~  ^^  +  -  =  ^^  +  ^  ^^ 

2  b         a  ab      ' 

11.  rf(3  X  -  9  c  +  14  6)  =  c{c  -  x). 

12.  (a  -b){x-  c)  -  (b  -c)(x-a)  =  (c-  a)(x  -  b). 

13.  (x  -  a)  (a  -b  -{-  c)  =  (x  +  a)  (b  -  a  +  c). 

14.  b{c  -  a:)  +  a{b  -  x)  -  b(b  -  a:)  =  0. 

^5    4  (3  -  2  a:)  X       ^      3 

m  —  n  rfi  —  rri^      vi  ■\-  n 

16.  a^x  +  &8:r  +  3  x{a%  +  ab'^)  =  3  ab. 

,«    4  a:  —  3a  ,  5  a:      K   .  15  6 

17. 1 =  O  -\ 

b  a  a 

T  Q    X  4-  a      X  +  c      X  +  b  _«i^_^c      J 


19. 


b  a  c         b      c      a 

ax  +  bx  -  a^  -  ah     2bx  -  2b^ -{- 2  ax  -  2  ab 


a  b 

20.  If  an  equation  is  an  identity  (cf.  §23),  how  many  roots  has  it? 
Tf  the  equation  ax  =  b  is  an  identity,  what  is  the  value  of  a?  Of  6? 
Of  the  root  6  h-  a  (cf.  §  55)  ?    Show  that  this  is  entirely  consistent. 


97-98]  SIMPLE  EQUATIONS  147 

II.   EQUATIONS   INVOLVING  FRACTIONS 

98.  Fractional  equations.  Equations  containing  expressions 
which  are  fractional  with  regard  to  an  unknown  number  (§  41) 
are  usually  called  fractional  equations.  Such  equations  frequently 
present  themselves  in  connection  with  practical  problems,  and  the 
process  of  solving  them  will  now  be  illustrated ;  the  demonstra- 
tion of  the  principles  involved  is  given  in  the  next  article. 

3      15       1 

Ex.  1.    Given  the  equation  ' = h  - ;  to  find  the  value  of  x. 

^  X      2      3a;      6' 

Solution.  If  each  member  of  the  given  equation  be  multiplied  by 
6x   (the  L.  CM.  of  the  denominators),  it  becomes 

18  -  3  a;  =  10  +  X, 

whence,  by  §  95,  x  =  2; 

moreover,  by  substituting  2  for  x,  it  is  found  that  the  given  equation  is 
satisfied,  hence  2  is  a  root  of  this  equation. 

Ex.2.    Given  ^ ^ =-^ ^^ ^;  to  find  a:. 

2  (a:  -  1)      7  (a:  +  1)      a:  +  1      7  (x^-  1) 

Solution.  On  multiplying  this  equation  by  2  •  7  •  (a:  +  1)  •  (a:  —  1), 
which  is  the  L.  C.  M.  of  the  denominators,  it  becomes 

3  .  7  .  (a;  +  1)  -  2  (a:  -  1)  =  8  .  2  •  7  .  (a:  -  1)  -  20, 

i.e.,  21  a; +  21 -2  a: +  2  =  112  a; -112-20, 

whence,  ^  =  f ; 

and,  on  being  substituted  for  x  in  the  given  equation,  |  proves  to  be  a 
root  of  that  equation. 

7      a:^  —  1 
Ex,  3.    Given   -  H =  a; ;  to  find  x. 

6      a:2  -  1 

Solution.     On  multiplying  this  equation  by  6(x^  —  1),  it  becomes 
7(a:2  _  1)  +  0(a:3  -  1)  =  6  a:(a:2  -  1), 
i.e.,  7x^-7-\-6x^-6  =  Qx^-Qx, 

whence,  7  a;^  +  6  x  -  13  =  0, 

i.e.,  (a;- l)(7a:  + 13)=0, 

and  the  roots  of  this  equation  are  1  and  —  ^p-v  [§  72 


148  ELEMENTARY  ALGEBliA  [Ch.  X 

But  by  trial  it  is  found  that  —  V  is  a  root  of  the  given  fractional  equa- 
tion, and  that  1  is  not  a  root  of  that  equation. 

Note.    Clearing  an  equation  of  fractions  mmj  bring  in  extraneous  roots,  i.e., 

roots  which  do  not  belong  to  the  given  equation;   this  is  illustrated  in  Ex.  3 

where  the  extraneous  root  1  was  brought  in  by  multiplying  by  the  unnecessary 

3.3 1 

factor  a;  —  1  in  clearing  the  given  equation  of   fractions ;    the   fraction   — 

might  first  have  been  reduced  to  its  lowest  terms. 

The  method  employed  for  solving  fractional  equations  in  the 
examples  given  above  may  be  stated  thus :  (1)  clear  the  given 
equation  of  fractions  hy  multiplying  it  hy  the  L.  C.  M.  of 
its  denominators,  (2)  solve  the  resulting  integral  equation, 
and  (3)  substitute  the  roots  of  this  integral  equation  in  the 
given  fractional  equation,  and  reject  those  ivhich  prove 
to  he  extraneous. 

EXERCISES 

Solve  the  following  equations : 

A        ^  X  _    5  __  ^  Q  3 2  n 

'364'  .  '  ^  ~    • 

5    a:-3a:-f5      ^  +  2^^ 
•       7  3  6  ■  ^+1  y 

10.  l-i  =  A_i. 

10      4.?/      by 


8. 

3 

X 

2 

-1 

9. 

y-^. 

=  1 

1 

6. 

X  — 

"2 

x,^. 

a:-3 
4 

J_ 

-5a; 
6 

7. 

4_ 

X 

13 
16' 

=  1  + 

8 

11.  _^+_^=_^+3. 

X^  -1       X  -1       x+1 

12.  Define  a  fractional  equation.  Are  the  equations  in  Exs.  4-6 
fractional?     Are  Exs.  7-11  fractional?     Why? 

13.  In  solving  the  equations  in  Exs.  4-6,  are  the  successive  equations 
equivalent?     Why?     Is  this  true  with  reference  to  Exs.  7-11  also? 

14.  If  in  Ex.  11  we  clear  of  fractions  and  simplify,  we  obtain  the 
equation  x^-2x-S  =  0,  i.e.,  (x -{■  1)  (x  -  Z)  =  0,  whose  roots  are  -1 
and  3.  Is  3  a  root  of  the  given  equation?  Is  -  1  a  root  (cf.  §  55)? 
Was  the  factor  x  +  1  necessary  to  clear  of  fractions?  Compare  also 
Ex,  3,  which  is  solved  above. 


98-99]  SIMPLE  EQUATIONS  149 

Solve  the  following  equations,  and  test  your  results : 

15    ^-1,^  +  2^  1  24    2x     (5^-3)      1^^ 

*  a;  -  2      x+1      x2  -  X  -  2  '3       10  x^  _  i      ^j 


7.2V-3  ./  2.(1-5)      3x(l-t) 

9^ 

z-\-  5       Z  +  10       2  +  4       2  + 


^^-   3-y      8"^2/  +  3       8(y  +  3)'      25.   IIAI^  +  IX-^A^^ZI. 

,„      2-5         2-10         2-4         2-9 


25.  ..  .  .  ,^ 


Suggestion.    Simplify  each  member  ^6 
before  cleariug  of  fractious. 


18. 


a:  +  1  _x  -\-  2  _  X  +  5  _  a:  +  6 
a;  +  2      a:  +  3'~x  +  6      a;  +  7 


19    -^  ~  1   I  3^  —  7  _  a:  —  5      a:  —  3 

■   a:-2a:-8~a;-6a;-4' 


20. 


21. 


a-3  +  2      a;8  -  2  _     10 


a;  +  l        a:—  1       x^  —  \ 

-  (2  -  a;)  -  -  (3  -  2  x)  =  ^^tl^- 
2^  ^4^  ^  6 


22.    -i-  +  ?=6 


23. 


1  —  a:      X  1  —  a; 

2  .x-  +  1  _        8        ^  2  a:-  1 
2a;-l      4a;2-l      2  x  +  l'  "    a{h-x)  '  h{c-x)     a{c-x) 


9(i 

,^     3     ,     18    21     ,     100     5 

17  +  -     1  +  — 1 +K 

X           XX         ,    ^      3 

3      '      5            9      '       15 

.. .  -'^-. 

27. 

-r-i 

28. 

2  c      6  _      c              2  ca; 

a       X      2-  X      a(2-x) 

29. 

=  a-b  + 

X  -\-  b                       X  +  a 

30. 

x^  —  ax  1  «  —  a:  _  Q 
a;2  +  ca:  —  az  —  «c      x  —  c 

31. 

a;  +  7rt  a:  — a_a:  +  7a  a  — x 
x  +  6a  '  X— 3a      x  +  a      2a  +  x 

32. 

./        i./     ,           '     ,=0. 

99.  Demonstration  of  principles  involved  in  §  98.  The  success 
of  the  method  employed  in  §  98  for  solving  fractional  equations 
is  due  to  the  fact  that,  in  the  great  majority  of  cases,  the  integral 
equation  obtained  by  clearing  an  equation  of  fractions  is  equiva- 
lent (§  95)  to  the  given  fractional  equation ;  the  exceptions,  as  the 
student  may  already  have  observed,  are  those  in  which  an  unneces- 
sary factor  is  used  to  clear  of  fractions  (cf.  Exs.  3  and  11,  §  98). 

To  prove  the  above,  let  it  first  be  recalled  that  transposing  and 
uniting  terms  (whether  those  terms  are  integral  or  fractional) 
leads  to  ah  equivalent  equation  (§  95).  Hence,  by  performing 
these  operations,  every  fractional  equation  may  be  reduced  to  an 
equivalent  equation  of  the  form 

f=0.  (1) 


150  ELEMENTARY  ALGEBRA  [Ch.  X 

wherein  JV  and  D  represent  integral  expressions  in  the  unknown 
number  (say  x),  and  D  is  the  L.  C.  M.  of  the  denominators  of  the 
fractions  in  the  given  equation. 

If  now  N  and  D  have  no  common  factor  (as  usually  happens), 
then  (§§  72  and  48)  there  is  no  value  of  x  for  which  both  N  and 
D  will  become  zero,  and  therefore  Eq.  (1)  is  equivalent  to 

N=0,  (2) 

i.e.,  the  given  equation  is  equivalent  to  Eq.  (2)  ;  but  Eq.  (2)  is  the 
result  of  clearing  the  given  equation  of  fractions,  hence,  in  all  such 
cases,  clearing  an  equation  of  fractions  leads  to  an  equivalent 
integral  equation. 

If,  on  the  other  hand,  N  and  D  have  a  common  factor — which 
rarely  happens  —  then  Eq.  (2)  is  not  equivalent  to  Eq.  (1);  for, 
if  N=F-N'  SLYid  D  =  F'  D',  where  F  is  the  H.  C.  F.  of  JST  and 
D,  then  only  thqse  values  of  x  for  which 

N'  =  0  (3) 

are  roots  of  Eq.  (1),  while  Eq.  (2)  has  all  of  these  roots,  and  also 

those  for  which  -n     /x  /is 

F  =  0;  (4) 

these  extraneous  roots  were  brought  in  by  using  the  unnecessary 
factor  F  to  clear  the  given  equation  of  fractions,  —  tJiey  are  those 
roots  of  Eq.  (2)  which  will  make  D  =  0,  and  are,  therefore,  easily 
detected. 

EXERCISES 

1.    Show  that  clearing  the  equation  — —  +       ^^       ~       ^^ 


x-7  x  +  2  9(x-\-2)  7{x-l) 
of  fractions,  by  the  usual  method,  produces  an  integral  equation  which 
is  equivalent  to  the  given  fractional  equation,  i.e.,  show  that  multiplying 
this  equation  by  the  L.  C.  M.  of  its  denominators  introduces  no  extrane- 
ous root. 

2.   Show  that  while  2  is  a  root  of  the  integral  equation  resulting  from 

clearing  — ^  + -3? =  8  +  — ^  of  fractions,  it  is  not  a  root 

a^  +  5      (r+5)(x-2)  x-2 

of  the  fractional  equation  itself.  What  is  the  value  of  D  (see  demon- 
stration above)  for  this  equation  when  a:  =  2  ?  How  may  extraneous 
roots  be  most  easily  detected? 


99]  SIMPLE  EQUATIONS  151 

Solve  the  following  equations,  and  tell,  by  mere  inspection,  i.e.,  without 
substituting  in  the  original  equation,  which  of  the  roots  of  the  integral 
equations,  if  any,  are  extraneous ;  also  state  your  reasons  in  full: 

3       20  40  ,  n        4:x 


3  a:  15  10 


_  -  5. 

x+1      'dx^  +  X-2     6x-2 

5        2  5x     ^  X  +  29 g 

'   x-5     3x  +  2      (x-5)(3x+2) 

c       12  a:         2a:  +  6_^ 


3  a;  —  7       a:  -  3 

PROBLEMS 

By  the  method  of  §  26  solve  the  following  problems,  applying  also  the 
principles  of  the  present  chapter: 

1.  If  I  of  a  certain  number  is  diminished  by  J  of  that  number,  and 
the  result  is  3  more  than  |  of  the  number,  what  is  the  number  ? 

2.  B's  present  age  is  18  years,  which  is  |  of  A's  age ;  after  how  many 
years  will  B's  age  be  f  of  A's  age  ? 

3.  The  tail  of  a  fish  is  4  inches  long.  Its  head  is  as  long  as  its  tail 
and  }  of  its  body,  and  its  body  is  as  long  as  the  head  and  ^  of  its  tail; 
how  long  is  its  body  ? 

4.  Mary,  who  is  now  24  years  old,  is  twice  as  old  as  Ann  was  when 
Mary  was  as  old  as  Ann  now  is.     How  old  is  Ann? 

5.  A  boy  bought  some  apples  for  24  cents ;  had  he  received  4  more 
for  the  same  sum,  the  cost  of  each  would  have  been  1  cent  less.  How 
many  did  he  buy  ? 

6.  A  reservoir  is  fitted  with  three  pipes,  one  of  which  can  empty  it  in 
4  hours,  another  in  3  hours,  and  the  third  in  1^  hours.  If  the  reservoir  is 
half  full,  and  the  three  pipes  are  opened,  in  what  time  will  it  be  emptied? 

7.  A  man's  age  is  such  that  |  of  it,  less  I  of  what  it  will  be  a  year 
hence,  is  equal  to  |  of  what  it  was  5  years  ago ;  how  old  is  he  ? 

8.  An  orchard  has  twice  as  many  trees  in  a  row  as  it  has  rows.  By 
increasing  the  number  of  trees  in  a  row  by  2,  and  the  number  of  rows 
by  3,  the  whole  number  of  trees  will  be  increased  by  126.  How  many 
trees  are  there  in  the  orchard  ? 


152  ELEMENTARY  ALGEBRA  [Ch.  X 

9.  Wliat  number  must  be  added  to  each  term  of  the  fraction  j'r  so 
that  the  resulting  fraction  shall  be  |  ? 

10.  If  a  certain  number  be  added  to,  and  also  subtracted  from,  each 
term  of  the  fraction  f,  the  first  result  will  exceed  the  second  by  |.  What 
is  the  number?    How  many  solutions  has  this  problem? 

11.  The  combined  cost  of  a  table  and  a  chair  is  ^11,  of  the  table  and 
a  picture,  $14,  and  the  chair  and  the  picture  together  cost  3  times  as 
much  as  the  table.     What  is  the  cost  of  each  ? 

12.  A  field  is  twice  as  long  as  it  is  wide,  and  increasing  its  length  by 
20  rods  and  its  width  by  30  rods,  increases  its  area  by  2200  square  rods. 
What  are  the  dimensions  of  this  field  ? 

13.  In  a  certain  quantity  of  gunpowder,  which  is  a  mixture  of  salt- 
peter, sulphur,  and  charcoal,  the  saltpeter  weighs  6  lb.  more  than  ^  the 
whole,  the  sulphur  5  lb.  less  than  ^  of  the  whole,  and  the  charcoal  3  lb. 
less  than  \  of  the  whole.  How  many  pounds  of  each  of  these  constituents 
are  contained  in  this  quantity  of  gunpowder  ? 

14.  An  officer  in  forming  his  men  into  a  solid  square,  with  a  certain 
number  on  a  side,  finds  that  he  has  49  men  left  over,  and  if  he  puts  1 
more  man  on  a  side  he  lacks  50  men  of  completing  the  square.  How 
many  men  has  he  ? 

15.  A  regiment  drawn  up  in  the  form  of  a  solid  square  lost  60  men  in 
battle,  and  when  the  men  were  rearranged  with  1  less  on  a  side,  there 
was  1  man  left  over.     How  many  men  were  there  in  this  regiment? 

16.  In  a  regiment  which  is  drawn  up  in  the  form  of  a  solid  square,  it 
is  found  that  the  number  of  men  in  the  outside  5  rows,  counted  all 
around,  is  -^-^  of  the  entire  regiment.  How  many  men  are  there  in  this 
regiment?  Has  the  equation  of  (his  problem  [cf.  §  26  (3)]  one  or  two 
solutions?    Is  each  also  a  solution  of  the  problem  itself? 

17.  A  boy  was  engaged  at  15  cents  a  day,  to  deliver  a  daily  paper  to 
those  of  its  subscribers  who  live  in  a  certain  part  o£  the  city,  with  the 
added  condition,  however,  that  he  was  to  forfeit  5  cents  for  every  day  he 
failed  to  perform  this  service;  at  the  end  of  60  days  he  received  $7. 
How  many  days  did  he  serve  ? 

18.  A  man  was  hired  for  30  days  on  the  following  terms  :  for  every 
day  he  worked  he  was  to  receive  $2.50  and  his*  board,  while  for  every 
day  he  was  idle  he  was  not  only  to  receive  nothing,  but  was  charged 
75  cents  for  his  board.  If  at  the  end  of  the  period  he  received  $49, 
how  many  days  did  he  work  ? 


99]  SIMPLE  EQUATIONS  153 

19.  A  steamer  can  sail  20  miles  an  hour  in  still  water.  If  it  can  sail 
72  miles  with  the  current  in  the  same  time  that  it  can  sail  48  miles  against 
the  current,  what  is  the  velocity  of  the  current  ? 

20.  A  steamer  now 'goes  5  miles  downstream  in  the  same  time  that  it 
takes  to  go  3  miles  upstream,  but  if  its  rate  each  way  is  diminished  by 
4  miles  an  hour,  its  downstream  rate  will  be  twice  its  upstream  rate. 
What  is  its  present  rate  in  each  direction? 

21.  A  man  rows  downstream  at  the  rate  of  6  miles  an  hour,  and 
returns  at  the  rate  of  3  miles  an  hour.  How  far  downstream  can  he  go 
and  return  if  he  has  9  hours  at  his  disposal  ?  At  what  fate  does  the 
stream  flow  ? 

22.  The  sum  of  two  numbers  is  18,  and  the  quotient  of  the  less 
divided  by  the  greater  is  equal  to  I.     What  are  the  numbers? 

23.  Divide  the  number  25  into  two  such  parts  that  the  square  of  the 
greater  part  exceeds  by  75  the  square  of  the  lesser  part. 

24.  Divide  72  into  four  parts,  such  that  if  the  first  part  be  divided  by 
2,  the  second  multiplied  by  2,  the  third  increased  by  2,  and  the  fourth 
diminished  by  2,  the  four  results  will  all  be  equal. 

25.  What  number  must  be  subtracted  from  each  of  the  four  numbers, 
20,  24,  16,  and  27,  so  that  the  product  of  the  first  two  remainders  shall 
equal  the  product  of  the  second  two? 

26.  Find  a  number  such  that  its  square  being  diminished  by  9,  and 
this  remainder  being  divided  by  10,  the  quotient  is  greater  by  3  than  the 
number  itself.     How  many  solutions  has  this  problem? 

27.  A  line  28  inches  long  is  divided  into  two  parts  of  which  the  length 
of  the  shorter  is  |  that  of  the  longer.     What  is  the  length  of  each  part? 

28.  An  automobile  runs  10  miles  an  hour  faster  than  a  bicycle,  and  it 
takes  the  automobile  6  hours  longer  to  run  255  miles  than  it  does  the 
bicycle  to  run  63  miles.  Find  the  rate  of  each.  How  many  solutions 
has  the  equation  of  this  problem  ?  Is  each  of  these  also  a  solution  of  the 
problem  itself? 

29.  At  what  time  between  2  and  3  o'clock  are  the  hands  of  a  clock 
together? 

Suggestion.  Let  z  represent  the  number  of  minute  spaces  over  which  the 
minute  hand  passes  from  2  o'clock  on,  until  it  overtakes  the  hour  hand  between 

2  and  3  o'clock,  then  show  that  :^+10  represents  the  same  number,  and  thus 

form  an  equation  and  find  the  value  of  z. 


154  ELEMENTARY  ALGEBRA  [Ch.  X 

30.  At  what  time  between  3  and  4  o'clock  is  the  minute  hand  15  minute 
spaces  ahead  of  the  hour  hand  ? 

31.  At  what  time  between  8  and  9  o'clock  are  the  hands  of  a  clock 
together  ?  « 

32.  At  what  time  between  4  and  5  o'clock  do  the  hands  of  a  watch 
extend  in  opposite  directions  ? 

33.  At  what  time  between  9  and  10  o'clock  is  the  hour  hand  20  minute 
spaces  in  advance  of  the  minute  hand? 

34.  In  an  alloy  of  silver  and  copper  weighing  90  oz.,  there  is  6  oz. 
of  copper;  find  how  much  silver  must  be  added  in  order  that  10  oz. 
of  the  new  alloy  shall  contain  but  |  of  an  ounce  of  copper. 

35.  If  80  lb.  of  sea  water  contains  4  lb.  of  salt,  how  much  fresh  water 
must  be  added  in  order  that  45  lb.  of  the  new  solution  may  contain  1|  lb. 
of  salt? 

36.  What  percentage  of  evaporation  must  take  place  from  a  6% 
solution  of  salt  and  water  (salt  water  of  which  6  %  by  weight  is  salt)  in 
order  that  the  remaining  portion  of  the  mixture  may  be  a  12%  solution? 
That  it  may  be  an  8  %  solution  ? 

37.  How  many  minutes  is  it  before  4  o'clock,  if  |  of  an  hour  ago  it 
was  twice  as  many  minutes  past  2  o'clock  ? 

38.  If  the  specific  gravity  of  brass  is  8|,*  while  that  of  iron  is  7^,  and 
if  an  alloy  of  brass  and  iron,  which  weighs  57  lb.,  displaces  7  lb.  of  water 
when  it  is  immersed,  what  is  the  weight  of  each  of  these  metals  in  the 
alloy? 

39.  If,  on  being  immersed  in  water,  97  oz.  of  gold  displaces  5  oz.  of 
water,  and  21  oz.  of  silver  displaces  2  oz.  of  water,  how  many  ounces  of 
gold  and  of  silver  are  there  in  an  alloy  of  these  metals  which  weighs 
320  oz.,  and  which  displaces  22  oz.  of  water?  What  is  the  specific  gravity 
of  each  of  these  metals  and  of  the  alloy  ? 

40.  Two  boat  builders,  A  and  B,  working  together,  can  build  a  boat 
of  a  certain  size  in  12  days,  and  A,  working  alone,  can  build  such  a 
boat  in  18  days.  In  how  many  days  can  B  alone  build  such  a  boat 
(cf.  Prob.  31,  §  26)  ? 

41.  A,  B,  and  C  together  can  do  a  piece  of  work  in  3^  days ;  B  can 
do  ^  as  much  as  A,  and  C  can  do  |  as  much  as  B.  In  how  many  days 
can  each  do  this  work  alone  ? 

*  This  means  that  a  given  volume  of  brass  weighs  8§  times  as  much  as  an  equal 
volume  of  water. 


99]  SIMPLE  EQUATIONS  156 

42.  A  can  do  a  certain  piece  of  work  in  6  days,  and  B  can  do  the  same 
work  in  14  days.  A,  having  begun  this  work,  had  later  to  abandon  it, 
when  B  took  his  place  and  finished  the  work  in  exactly  10  days  from  the 
time  it  was  begun  by  A.     How  many  days  did  B  work  at  it  ? 

43.  A  and  B  can  dig  a  certain  trench  in  10  days,  B  and  C  can  dig  it  in 
6  days,  and  A  and  C  in  7^  days.  How  long  would  it  take  each  working 
alone  to  do  this  work  ? 

44.  The  first  of  three  outlet  pipes  can  empty  a  certain  cistern  in  2  hr. 
and  40  min.,  the  second  in  3  hr.  and  15  min.,  and  the  third  in  4  hr.  and 
25  min.  If  the  cistern  is  |  full,  and  all  three  pipes  are  opened  at  the  same 
time,  how  long  will  it  take  to  empty  it  ? 

45.  A  gentleman  invested  \  of  his  capital  in  4%  bonds,*  f  of  it  in 
3^7o  bonds,  and  the  remainder  in  6%  bonds,  purchasing  all  these  bonds  at 
par.  If  his  total  annual  income  is  $2100,  what  is  the  amount  of  his 
capital  ? 

46.  A  gentleman  made  two  investments  amounting  together  to  |4330 ; 
on  the  first  he  lost  5%  and  on  the  second  he  gained  12%.  If  his  net  gain 
was  $251,  what  was  the  amount  of  each  investment? 

47.  An  estate  was  divided  among  four  heirs.  A,  B,  C,  and  D.      The 

amounts  received  by  A  and  B  were,  respectively,  |  and  \  of  an  amount 
$  1000  less  than  the  estate ;  and  C  and  D  received,  respectively,  \  and  J- 
of  an  amount  greater  than  the  estate  by  ^  of  it.  How  much  did  each 
receive  ? 

48.  A  wheelman  and  a  pedestrian  start  at  the  same  "time  for  a  place 
54  miles  distant,  the  former  going  3  times  as  fast  as  the  latter ;  the  wheel- 
man, after  reaching  the  given  place,  returns  and  meets  the  pedestrian 
6f  hours  from  the  time  they  started.     At  what  rate  did  each  travel  ? 

49.  A  girl  found  that  she  could  buy  12  apples  with  her  money  and 
have  5  cents  left,  or  10  oranges  and  have  6  cents  left,  or  6  apples  and  6 
oranges  and  have  2  cents  left.     How  much  money  had  she? 

50.  Find  a  fraction  whose  numerator  is  greater  by  3  than  one  half  of 
its  denominator,  and  whose  value  is  |. 

51.  The  numerator  of  a  certain  fraction  is  less  by  8  than  its  denomi- 
nator, and. if  each  of  its  terms  be  decreased  by  5,  its  value  will  be  ^;  what 
is  the  fraction  ? 

52.  The  tens'  digit  of  a  certain  two-digit  number  is  \  the  units'  digit, 
and  if  this  number,  increased  by  27,  be  divided  by  the  sum  of  its  digits, 
the  quotient  will  be  6J.     What  is  the  number  (cf.  Prob.  4,  §  26)  ? 

*  Bonds  yielding  4%  interest  per  annum. 


156  ELEMENTARY  ALGEBRA  [Ch.  X 

53.  A  certain  number  is  increased  by  1,  and  also  diminished  by  1,  and 
it  is  then  found  that  3  times  the  reciprocal  of  the  first  result,  being 
increased  by  I,  equals  2  times  the  reciprocal  of  the  second.  What  is  this 
number?     How  many  solutions  has  this  problem? 

54.  A  steamer's  speed  is  such  that,  on  a  certain  stream,  it  takes  as  long 
to  go  3  miles  upstream  as  it  does  t6  go  5  miles  downstream,  L)ut  if  its 
rate  in  still  water  were  4  miles  less  per  hour,  its  downstream  rate  would 
be  twice  its  upstream  rate.    What  is  its  rate  in  still  water? 

55.  A  physician  having  a  6%  solution  of  a  certain  kind  of  medicine 
wishes  to  dilute  it  to  a  3^  %  solution.  What  percentage  of  water  must 
he  add  to  the  present  mixture  ? 

56.  A  physician  having  a  6%  solution,  and  also  a  3%  solution,  of  a 
certain  kind  of  medicine,  mixes  these  in  such  proportions  as  to  form  a 
3^%  solution.  What  percentage  of  the  new  mixture  is  taken  from  each 
of  the  given  mixtures  ? 

57.  A  tourist  ascends  a  certain  mountain  at  an  average  rate  of  1^  miles 
an  hour,  and  descends  by  the  same  path  at  an  average  rate  of  4J  miles  an 
hour.  If  it  takes  him  6|  hours  to  make  the  round  trip,  how  long  is  the 
path  ? 

58.  If  a  father  takes  3  steps  while  his  son  takes  5,  and  if  2  of  the 
father's  steps  are  equal  in  length  to  3  of  the  son's,  how  many  steps  will 
the  son  have  to  take  before  he  overtakes  his  father,  who  is  36  of  his  own 
steps  ahead  ? 

Solution,  The  simplest  way  to  form  the  equation  of  this  problem  is  to  com- 
pare two  lengths.     To  do  this 

let  I  =  the  number  of  feet  in  the  son's  step, 

3  I 
then  —  =  the  number  of  feet  in  the  father's  step ; 

also  let  X  =  the  number  of  steps  the  son  must  take, 

then  —  =  the  number  of  steps  the  father  will  take ; 

5 

and  the  equation  of  the  problem  is 

a^^=^-Y  +  3G.-^,  (why?) 

i.e.,  a;  =  ^  +  54 ;  whence  a;  =  540. 

Observe  that  fractions  could  have  been  avoided  by  letting  5  x  and  2 1,  respec- 
tively, stand  for  the  number  and  length  of  the  son's  steps. 


99-100]  SIMPLE  EQUATIONS  157 

59.  A  hare  pursued  by  a  hound  takes  4  leaps  while  the  hound  takes  3, 
but  2  of  the  hound's  leaps  are  equal  in  length  to  3  of  the  hare's.  If 
the  hare  has  a  start  equal  to  60  of  her  own  leaps,  how  many  leaps 
must  the  hound  take  to  catch  the  hare? 

60.  Solve  Prob.  59  if  all  its  conditions  are  unchanged  except  that  the 
hare's  start  is  equal  to  60  of  the  hound's  leaps. 

61.  A  merchant  added  annually  to  his  capital  an  amount  equal  to  ^  of 
it,  but  deducted  at  the  end  of  each  year  $2000  for  personal  expenses.  If 
after  taking  out  the  1 2000  at  the  end  of  the  third  year,  he  finds  that  he 
has  just  twice  his  original  capital,  what  was  the  original  capital  ? 

62.  A  pedestrian  finds  that  his  uphill  rate  of  walking  is  3  miles  an 
hour,  while  his  downhill  rate  is  4  miles  an  hour.  If  he  walked  60  miles 
in  17  hours,  how  much  of  this  distance  was  uphill? 

63.  A  hound  is  39  of  his  leaps  behind  a  rabbit  that  takes  7  leaps 
while  the  hound  takes  8.  If  6  leaps  of  the  rabbit  are  equal  to  5  leaps  of 
the  hound,  liow  many  leaps  must  the  hound  take  to  catch  the  rabbit? 

64.  A  picture  whose  length  lacks  2  inches  of  being  twice  its  width, 
is  inclosed  in  a  frame  4  inches  wide.  If  the  length  of  the  frame  divided 
by  its  width,  plus  the  length  of  the  picture  divided  by  its  width,  is  3J, 
what  are  the  dimensions  of  the  picture?  How  many  solutions  has  the 
equation  of  this  problem  ?    Is  each  of  these  a  solution  of  the  problem  also  ? 

III.     GENERAL  PROBLEMS 

100.  General  problems.  Interpretations  of  their  solutions.  A 
problem  in  which  the  known  numbers  are  represented  by  letters, 
instead  of  by  the  Arabic  characters,  is  often  called  a  general 
problem,  because  it  includes  all  those  particular  problems  which 
may  be  obtained  by  giving  particular  values  to  these  letters — com- 
pare §  9,  and  also  the  illustrations  given  below. 

Prob.  1.  A  yacht  was  chartered  for  a  pleasure  party  consisting  of 
p  persons,  the  expense  to  be  shared  equally  by  those  participating;  q 
of  the  proposed  party  being  unable  to  go,  it  was  found  necessary  for 
each  person  who  did  go  to  pay  d  dollars  more  than  would  otherwise 
have  been  necessary.  How  much  was  paid  for  the  yacht?  How  much 
was  each  to  pay  under  the  original  ari-angement ? 

Solution 
Let  X  =  the  number  of  dollars  each  member  of  the  original  party  was 
to  have  paid,  then  x  +  d  is  the  number  of  dollars  that  each  participant 


158  ELEMENTARY  ALGEBRA  [Ch.  X 

actually  did  pay,  while  px  and  (p  -  q)  ■  (x  +  d)  are  two  expressions,  each 
of  which  represents  the  number  of  dollars  charged  for  the  yacht ;  hence 

px  =  {p  —  q)  (x  +  d)  =  px  +  pd  —  qx  —  qd\ 
whence  x  =    v^  ~  ^^^  the  amount  each  was  to  pay, 

and  px  =  p  ■  -^ — -^,  the  price  of  the  yacht. 

The  student  may  solve  this  problem  independently  if  p  =  12,  q  =  3, 
and  d  =  2,  and  compare  the  results  with  those  obtained  by  substituting 
these  values  for  p,  q,  and  d  in  the  above  general  solution  (formula). 

Prob.  2.  Divide  m  golf  balls  into  two  groups  in  such  a  way  that  the 
first  group  shall  contain  n  balls  more  than  the  second  group. 

Solution 
Let  X  =  the  number  of  balls  in  the  first  group. 

Then  m  —  x  =  the  number  of  balls  in  the  second  group, 

and,  therefore,  by  the  condition  of  the  problem, 
x  =  m  —  x-\-n; 

whence  x  =  ^  ^  ^,  the  number  of  balls  in  the  first 

group,  and  m  —  x  =  m  —  ^  "^  ^  =  ^  ~  ^,  the  number  of  balls 

in  the  second  group. 

As  in  the  previous  problem,  so  here,  the  general  solution  just  obtained 
may  be  employed  to  obtain  the  solution  of  any  particular  problem  of 
the  same  kind.     For  example,  if  7n  =  30  and  n  =  4,  then  the  two  groups 

contain,  respectively,  — - —  and  — - —  balls,  i.e.,  17  and  13 ;  while,  if 

771  =  10  and  n  =  2,  then  the  two  groups  contain  6  and  4  balls,  respectively. 
If,  however,  m  =  10  and  n  =  14,  then  the  number  of  balls  in  the  two 

groups,  as  given  by  the  above  solution,  is  — '^ —  and  — p — ,  respec- 
tively, i.e.,  12  and  —  2 ;  but  since  there  can  not  be  an  actual  group  con- 
taining —  2  golf  balls,  therefore  this  last  problem  is  impossible,  and  the 
impossibility  is  indicated  by  the  negative  result. 

Note  1.  Some  problems  admit  of  negative  results,  and  some  do  not,  just  as 
some  problems  admit  of  fractional  results,  while  others  do  not.  The  nature  of 
the  things  with  which  any  particular  problem  is  concerned  will  always  make 
it  evident  whether  or  not  fractional  or  negative  solutions  are  admissible. 


100]  SIMPLE  EQUATIONS  159 

For  example,  let  it  be  required  to  find  the  temperature  at  Chicago  on  a  certain 
day,  it  being  known  that  on  that  day  the  sum  of  the  thermometer  readings  at 
New  York  and  Chicago  is  10°,  their  difference  14°,  and  that  it  is  colder  in  Chicago 
than  in  New  York. 

Let  the  reading  at  Chicago  be  x  degrees.  Then  it  is  (10  —  x)  degrees  at  New 
York,  and  the  other  condition  of  the  problem  becomes  x—  (10  — x)  — 14,  whence 
x=— 2,  i.e.,  the  reading  at  Chicago  is  2°  below  zero.  The  negative  result  is 
admissible  in  this  problem. 

Note  2.  Observe  also  that  two  algebraic  problems  which  differ  widely  with 
regard  to  the  tilings  with  which  the  problems  are  concerned  may  yet  give  rise 
to  the  same  equation,  and  the  solution  of  this  equation  may  be  a  solution  of 
one  of  the  problems,  while  it  merely  shows  that  the  other  problem  demands  what 
is  impossible  of  fulfilment. 

Thus,  if  the  head  of  a  certain  fish  is  7|  inches  long,  the  tail  as  long  as  the 
head  and  J  of  the  body,  and  the  body  as  long  as  the  head  and  tail  together,  how 
long  is  the  body  of  the  fish  ? 

If  X  =  number  of  inches  in  the  length  of  the  body,  then  the  second  condition 

of  the  problem  becomes  x  =  7|4-7j  +  -,  i.e.,  x  =  15  +  ^,  whence  x  =  22^. 

o  o 

This  number  is  found  to  satisfy  all  the  conditions  of  the  given  problem,  and 
is,  therefore,  not  only  the  solution  of  the  equation,  but  is  also  the  solution  of 
the  problem. 

Again,  let  it  be  required  to  find  how  many  sparrows  a  certain  dock  must  con- 
tain if  j^2  of  their  number,  plus  ^  of  their  number,  plus  15,  equals  the  whole 
number. 

If  X  =  their  number,  then  the  given  condition  becomes  x  =  :^  +  -  +  15,  i.e., 

x=  15  +  -,  which  is  the  same  as  the  equation  in  the  former  problem,  but  the 

solution  of  this  equation,  viz.,  x  =  22^,  is  not  now  a  solution  of  the  problem, 
but  merely  shows  the  impossibility  of  fulfilling  the  conditions  of  the  problem. 

Prob.  3.  Two  boys,  A  and  B,  are  running  along  the  same  road,  A  at 
the  rate  of  a  rods  per  minute,  and  B  at  the  rate  of  b  rods  per  minute ; 
if  B  is  m  rods  in  advance  of  A,  and  if  they  continue  running  at  the  same 
rates,  in  how  many  minutes  v\rill  A  overtake  B  ? 

Solution 
Let  X  =  the  number  of  minutes  that  must  elapse  before  A  overtakes  B. 
Then,  by  the  conditions  of  the  problem, 

ax  =  bx  -{■  m, 

whence  x  = ,  the  number  of  minutes  before 

A  overtakes  B.  ^  ~ 

As  in  the  two  previous  problems,  so  here  the  general  solution  just  obtained 
may  be  employed  to  find  the  solution  of  any  particular  problem  of  the  same  kind. 

00 

E.g.,  if  a  =  60,  6  =  50,  and  m  =  90,  then  x  =  =  9,  i.e.,  A  will  overtake  B 

in  9  minutes.  60     50 


160  ELEMENTARY  ALGEBRA  [Ch.  X 

Again,  if  a  =  50,  6  =  50,  and  m  =  90,  then  x=— ^— —  =  — ,  i.e.,  an  infinite 

50  —  50       0 
number  of  minutes  will  elapse  before  A  overtakes  B ;  in  other  words,  A  will  never 
overtake  B.     Compare  §  55,  and  also  the  note  to  Ex.  15,  of  §  55. 

CM) 

But  if  a  =  50,  6  =  GO,  and  m  =  90,  then  x  =  — = =  —  9,  i.e.,  the  two  boys  are 

50  —  ()0 
together  —  9  minutes  from  the  moment  they  were  observed,  and  since  adding  —  9 
minutes  to  the  present  time  is  the  same  as  subtracting  9  minutes  from  the  present 
time,  therefore  the  two  boys  were  together  9  minutes  ago. 

This  interpretation  of  the  negative  result  accords  fully  with  the  physical  condi- 
tions of  the  actual  problem,  because  if  B  is  already  90  rods  in  advance  of  A,  and 
if  he  is  running  10  rods  per  minute  faster  than  A,  he  will  not  only  keep  getting 
farther  and  farther  ahead  of  A,  but  he  must  also  have  passed  him  9  minutes  ago. 

Prob.  4.    The  present  ages  of  a  father  and  son  are  respectively  50 

and  20  years ;  after  how  many  years  will  the  father  be  4  times  as  old  as 

the  son  Y 

Solution 

Let  X  =  the  number  of  years  from  now  to  the  time  when  the  father's 
age  shall  be  4  times  that  of  the  son.  Then,  by  the  conditions  of  the 
V^ohlem,  50  +  x  =  4(20  +  :r), 

whence  a;  =  —  10. 

This  means  that  10  years  ago  the  age  of  the  father  was  4  times  that  of 
the  son. 

N.  B.  The  general  problem,  which  includes  Prob.  4  as  a  particular  case,  may 
be  stated  thus :  The  present  ages  of  a  father  and  son  are,  respectively,  m  and  n 
years ;  after  how  many  years  will  the  father's  age  be  p  times  that  of  the  son  ? 

EXERCISES  AND  PROBLEMS 

5.  Is  Prob.  25  of  §  99  a  particular  or  a  general  problem?  Why? 
Formulate  a  general  problem  which  shall  include  this  one  as  a  particular 
case.  Solve  the  new  problem  and  thus  find  a  formula  by  which  Prob.  25 
may  be  solved. 

6.  Answer  the  questions  in  Ex.  5  above,  supposing  them  to  have  been 
asked  wath  regard  to  Prob.  24,  p.  153. 

7.  Answer  the  questions  in  Ex.  5  above,  supposing  them  to  have  been 
asked  with  regard  to  Prob.  10,  p.  152. 

8.  Does  Prob.  24,  p.  153,  admit  of  a  fractional  result?  Of  a  negative 
result  ?     Explain  your  answers. 

9.  By  a  slight  change  in  the  wording  of  Prob.  4,  §  100,  make  an 
equivalent  problem  of  which  the  answer  shall  be  positive.  This  should 
agree  with  the  interpretation  there  given  of  the  negative  result. 


100]  SIMPLE  EQUATIONS  161 

10.  By  slightly  changing  the  wording  of  the  last  particular  case  under 
Prob.  3,  §  100,  make  an  equivalent  problem  whose  answer  shall  be  positive. 

11.  A  farmer  can  plow  a  certain  field  in  a  days,  and  his  son  can  plow 
the  same  field  in  (f  days.  In  how  many  days  can  both  working  together 
plow  the  field  ? 

12.  Is  Prob.  11  a  particular  or  a  general  problem?  Make  several 
examples  of  which  it  is  the  generalization.  Solve  one  of  these  particular 
examples  independently,  and  then  show  that  its  answer  could  have  been 
obtained  from  the  answer  to  Prob.  11. 

13.  At  what  time  between  n  and  n  + 1  o'clock  will  the  hands  of  a  clock 
be  together?  At  what  time  between  these  hours  will  they  be  pointing  in 
opposite  directions,  ifn<6?    Ifn>6?     Ifn  =  6? 

14.  A  fathei-  is  m  times  as  old  as  his  son,  and  in  p  years  he  will  be  n 
times  as  old.     Find  their  respective  ages. 

Interpret  your  result  when  m<,n.  Is  jo  positive  or  negative  in  this 
case? 

15.  A  merchant  has  two  kinds  of  sugar  worth,  respectively,  a  and 
h  cents  a  pound.  How  many  pounds  of  each  kind  must  be  taken  to  make 
a  mixture  of  n  pounds  worth  c  cents  a  pound? 

Interpret  the  result  if  a  =  ft,  and  c  is  less  than  a ;  also  when  a  =  h  =  c. 
Do  these  interpretations  of  the  results  agree  with  the  conditions  of  the 
problem  under  the  same  suppositions  ? 

16.  An  alloy  of  two  metals  is  composed  of  m  parts  (by  weight)  of  one 
to  n  parts  of  the  other.  How  many  pounds  of  each  of  the  metals  are 
there  in  a  pounds  of  the  alloy? 

Show  that  the  problem  just  stated  is  the  generalization  of  such  a  prob- 
lem as  this:  Bell  metal  is  an  alloy  of  5  parts  (by  weight)  of  tin  to  IG  of 
copper;  how  many  pounds  of  tin  and  of  copper  in  a  bell  weighing  4200  lb.? 

17.  A  wheelman  sets  out  from  a  certain  place  at  m  miles  an  hour,  and 
is  pursued  by  a  second  wheelman,  who  starts  from  the  same  place  a  hours 
later,  and  rides  p  miles  an  hour.  How  far  from  the  starting  point  will 
the  second  wheelman  overtake  the  first?  What  does  this  result  become 
if  m  =  10,  jt?  =  12,  and  a  =  4? 

18.  Two  wheelmen,  A  and  B,  are  observed  passing  a  certain  point,  A 
being  c  hours  in  advance  of  B,  and  traveling  at  the  rate  of  a  miles  in 
h  hours,  while  B  travels  p  miles  in  q  hours.  How  far  will  A  travel  before 
he  is  overtaken  by  B  ? 

Under  what  conditions  is  this  solution  positive?  Negative?  Zero? 
Infinite  ?    Interpret  the  result  in  each  case. 


CHAPTER   XI 

SIMULTANEOUS   SIMPLE   EQUATIONS 

I.   TWO  UNKNOWN   NUMBERS 

101.  Indeterminate  equations.  Although  a  simple  equation  in 
one  unknown  number  has  one  and  but  one  solution  (of.  §  97),  yet 
it  is  easy  to  see  that  an  equation  which  involves  two  or  more 
unknown  numbers  has  an  infinite  number  of  solutions. 

E.g.f  in  the  equation  x  +  3  y  =  5,  which  is  equivalent  to 

y  =  ^,  [§95 

there  is  a  perfectly  definite  value  of  y  corresponding  to  every  value  that  one  may- 
choose  to  assign  to  x ;  thus,  if  x  =  l,  then  ?/  =  f ,  if  a;  =  2,  ?/  =  1,  if  a;  =  3,  ?/  =  f ,  if 
x=—l,  y  =  2,  and  so  on  indefinitely;  i.e.,  each  of  these  pairs  of  numhers,  viz., 
1  and  f ,  2  and  1,  3  and  f ,  etc.,  constitutes  a  solution  of  the  given  equation,  because, 
when  substituted  for  x  and  y  respectively,  they  satisfy  that  equation. 

An  equation,  such  as  the  one  just  now  considered,  which  has 
an  infinite  number  of  solutions,  is,  for  that  reason,  called  an 
indeterminate  equation. 

102.  Positive  integral  solutions  of  indeterminate  equations.  Al- 
though the  number  of  solutions  of  an  indeterminate  equation,  as 
has  just  been  illustrated,  is  unlimited,  yet  it  often  happens  that 
only  solutions  of  a  particular  kind  are  sought,  —  e.g.,  those  that 
are  positive  integers,  —  and  the  number  of  these  may  be  finite. 

In  practice  the  positive  integral  solutions  of  an  indeterminate 
equation  can  usually  be  found  by  mere  inspection,  or  by  trial. 

E.g.,  to  find  the  positive  integral  solutions  of  the  equation  2  a;  +  3  y  =  7,  it  is 
only  necessary  to  assign  to  one  of  the  unknown  numbers,  say  x,  the  values  1,  2, 
3,  •••  in  turn,  and  to  find  the  corresponding  values  of  the  other  unknown  number, 
which  are  f,  1,  ^,  ••• ;  moreover,  if  a;  =  4,  or.  any  greater  number,  then  y  is  nega- 
tive, hence  the  only  positive  integral  solution  of  the  given  equation  is  x  =  2  and 
y  =  l. 

162 


101-103]  SIMULTANEOUS   SIMPLE  EQUATIONS  163 

Many  problems  lead  to  indeterminate  equations  which,  from  the 
nature  of  the  things  involved,  demand  solutions  that  are  positive 
integers. 

E.g.,  a  farmer  spent  S22  purchasing  two  kinds  of  lambs,  the  first  kind  costing 
him  $  3  each,  and  the  second  kind  $  5  each.    How  many  of  each  kind  did  he  buy  ? 

Solution.    Let  x  =  the  number  of  the  first  kind, 

and  y  =  the  number  of  the  second  kind. 

Then  one  condition  of  the  problem  is  that 

Sx+5tj  =  22, 
and  the  other  condition  is  that  z  and  y  shall  be  positive  integers.* 

By  §  95,  this  equation  is  equivalent  to  a;= — ^, 

o 

and,  if  the  values  1, 2,  3,  and  4  be  assigned  to  y,  the  corresponding  values  of  z  are 
found  to  be  V>  4,  I,  and  |;  moreover,  if  y  =  5,  or  more,  then  z  is  negative,  and 
therefore  the  ojily  positive  integral  solution  of  the  above  equation  is  a;  =  4  and 
y  =  2;  i.e.,  4  and  2  are,  respectively,  the  numbers  of  lambs  purchased. 

103.  Positive  integral  solutions :  another  method.  Another 
method  of  finding  the  positive  integral  solutions  of  an  indetermi- 
nate equation  will  now  be  illustrated. 

Given  the  equation  7 x  +4  ?/  =  46 ;  to  find  its  positive  integral  solutions. 
By  transposing  and  dividipg,  this  equation  becomes 

4  4 

i.e.,  y-ll-^z  =  ^^, 

and,  since  z  and  y  are  integers,  therefore  the  first  member  of  this  equation  repre- 

2 3  2; 

sents  an  integer,  and  therefore  the  second  member,  viz.,  — - — ,  also  represents 

an  integer. 

Again,  since  = represents  an  integer,  therefore  the  product  obtained  by 

4 

multiplying  it  by  any  integer  whatever  also  represents  an  integer ;  moreover,  if 
this  multiplier  be  so  chosen  that  the  new  coefiScient  of  z  shall  exceed  some  multi- 
ple of  the  denominator  by  1  (cf.  §  79),  then  the  integral  values  of  z  and  y  may  be 
easily  determined  as  follows : 

2 3  -J.  3/2 3  2;)      5 9  X  2 z 

Since  — : —  represents  an  integer,  therefore  -^ — - — ^  =  — 7 —  =  1—2  z-] — — 
4  2—x  4  4  4 

represents  an  integer,  and  therefore  represents  an  integer.    If  this  last 

2 X 

integer  be  designated  by  p,  then       — —  =p, 

-  *  Although  this  condition  is  not  expressible  by  means  of  an  equation,  yet  it  is 
none  the  less  vital  on  that  account. 


164  ELEMENTARY  ALGEBRA  [Ch.  XI 

whence  cc  =  2  —  4  p, 

and,  on  substituting  this  value  of  z  in  the  given  equation,  it  becomes 

y  =  8  +  7p. 

In  these  last  two  equations  x  and  ?/  are  positive  integers,  and  p  is  an  integer, 
though  not  necessarily  positive.  This  shows  that  p  is  either  —  1  or  0  (in  order 
that  X  and  y  may  be  positive),  whence  x  =  6  and  y  =  1,  ov  x  =  2  and  y  =  S;  and 
there  are  no  other  positive  integral  values  of  x  and  y  which  satisfy  the  given 
equation. 

EXERCISES 

Find  five  solutions  to  each  of  the  following  equations : 

1.   Sx-4y=8.  2.   2w  =  5  +  3z.  3.   3r  +  6s  =  20. 

4.  How  many  solutions  has  each  of  the  above  equations?     Why? 
What  are  such  equations  called? 

5.  If  possible  solve  the  equations  in  Exs.  1,  2,  and  3  above,  in  posi- 
tive integers.     How  many  such  solutions  has  each? 

Find  the  positive  integral  solutions  of  the  following  equations : 

6.  ?  +  ^  =  5.  7.   6x  +  7y  =  52.  8.    13  w  +  5  y  =  229. 

Show  that  the  following  equations  have  no  positive  integral  solutions  : 
9.  2x  -4:y  =  l.        10.   dx  +  6y  =  5.  11.   9  x  +  3  y  =  17. 

12.  Sliow  that  the  indeterminate  equation  ax  +  ly  =  c  can  not  be 
solved  in  positive  integers  when  a  +  ft  >  c ;  nor  when  a  and  h  have  a 
common  factor  which  is  not  a  factor  of  c. 

13.  Find  three  solutions  of  the  equation  2a:  —  5?/  +  32  =  6. 

14.  If  a  man  spends  $300  for  cows  and  sheep,  which  cost  respectively 
$  45  and  $  6  a  head,  how  many  of  each  does  he  purchase  ? 

15.  In  how  many  and  what  ways  may  a  19-pound  package  be  weighed 
with  5-pound  and  2-pound  weights  ? 

16.  How  many  pineapples,  at  25  cents  each,  and  watermelons,  at  15 
cents  each,  can  be  purchased  for  $2.15? 

17.  Divide  a  line  which  is  100  feet  long  into  two  parts,  one  of  which 
shall  be  a  multiple  of  11,  and  the  other  of  6. 

18.  Find  the  least  number  which  when  divided  by  4  gives  a  remainder 
of  3,  but  when  divided  by  5  gives  a  remainder  of  4. 

19.  A  man  selling  eggs  to  a  grocer  counted  them  out  of  his  basket  4 
at  a  time  and  had  1  egg  left  over,  and  the  grocer  counted  them  into  his 
box  5  at  a  time  and  there  were  3  left  over.  If  the  man  had  between 
()  and  7  dozen  eggs,  how  many  must  there  have  been  ? 


103-105]  SIMULTANEOUS   SIMPLE  EQUATIONS  165 

104.  Definitions:  simultaneous  equations,  etc.  Although  a  single 
equation  which  involves  two  unknown  numbers  has  just  been 
shown  to  be  indeterminate,  i.e.,  to  have  an  indefinite  number  of 
solutions,  yet  if  two  such  simple  equations  be  given,  it  usually 
happens  that  one,  and  only  one,  pair  of  numbers  can  be  found 
which  will  satisfy  each  of  them,  i.e.,  be  a  solution  of  each. 

E.g.,  the  equations  4  a;  +  3  y  =  5  and  2  x  —  5  y  =  9  are  each  satisfied  by  x  =  2 
andy  =—1,  and  by  no  other  pair  of  numbers. 

Two  or  more  equations  which  are  satisfied  by  the  same  set  (or 
sets)  of  numbers  are  called  simultaneous  equations  (also  called 
consistent  equations),  while  two  equations  which  have  no  solu- 
tion whatever  in  common  are  called  inconsistent  equations  (also 
called  incompatible  equations) ;  e.g.,  x  +  y  =  4:  and  2  x-\-2y  =  9  are 
inconsistent  equations. 

Two  or  more  equations  which  express  different  relations  be- 
tween the  unknow^n  numbers,  and  therefore  can  not  be  reduced 
to  the  same  form,  are  called  independent  equations. 

Two  or  more  equations  taken  together  are  often  called  a  system 
of  equations ;  and  any  set  of  numbers  which  satisfies  every  equa- 
tion of  the  system  is  called  a  solution  of  the  system. 

105.  Solving  simultaneous  equations.  The  process  of  finding  a 
solution  of  a  system  of  simultaneous  equations  is  called  solving  the 
equations;  this  process  will  now  be  illustrated  by  some  easy 
examples. 

f  X  +    7/  ■=    4:,  (1) 

Ex.  1.    Solve  the  equations  -; 

^  lx->j  =  2.  (2) 

Solution.     Adding  these  two  equations,  member  to  member,  gives 

2  a:  =6, 

whence  x  =  d. 

Substituting  this  value  of  x  in  Eq.  (1)  gives 

3  +  2/  =  4, 

whence  2/  =  !• 

That  these  numbers,  viz.,  x  =  S  and  ?/  =  1,  really  constitute  a  solution 
of  the  given  equations  is  verified  by  substituting  them  for  x  and  y  in 
those  equations. 


166  ELEMENTARY  ALGEBRA  [Ch.  XI 

rdx-^y  =  l,  (1) 

Ex.  2.    Solve  the  equations    -  x         ^ 

I     a; +  2?/ =  9.  .  (2) 

Solution.     On  multiplying  Eq.  (2)  by  2,  it  becomes 

2a:  +  42/  =  18,  (3) 

and  adding  Eq.  (3)  to  Eq.  (1)  gives 

5  a;  =  25, 

whence  x=  o\ 

and  the  corresponding  value  of  y  may  be  found  by  substituting  this  value 
of  X  in  either  of  the  equations  which  contain  both  x  and  y.  E.g.,  by  this 
substitution  Eq.  (2)  becomes 

5+2y  =  9, 

whence  ^  =  2 ; 

and  it  is  easily  verified  as  in  Ex.  1  that  x  =  5  and  y  =  2  is  a  solution  of 
each  of  the  given  equations. 

r  3  X  +  2  ?/  =  26,  (1) 

Ex.  3.   Solve  the  equations  -{ 

^  1 5  a: +  9  2/ =  83.  (2) 

Solution.     On  multiplying  both  members  of  Eq.  (1)  by  5,  and  of 
Eq.  (2)  by  3,  they  become,  respectively, 

15  a;  +  10  2/ =  130,  (3) 

15^  +  27  3/ =  249;  (4) 

and  subtracting  Eq.  (3)  from  Eq.  (4)  gives 

17?/ =119, 

whence  y  —  'J. 

Substituting  this  value  of  y  in  any  one  of  the  equations  containing 
both  X  and  y  gives  _  . 


it  isTfeai 


and  it  isTfea^ily  verified  that  x  =  4  and  ?/  =  7  is  a  solution  of  the  given 
system  6i^eqyations. 

Observl  tmxt  if  Eq.  (1)  had  been  multiplied  by  9,  and  Eq.  (2)  by  2, 
and  if  one  of  the  two  resulting  equations  liad  been  subtracted  from  the 
other,  then  y  would  have  disappeared,  and  the  value  of  x  would  have 
been  found  before  that  of  y. 


•(   />- 


105-106]  SIMULTANEOUS   SIMPLE  EQUATIONS  167 

Ex.  4.   Solve  the  equations 


V-ii=-|,  (1) 


f  +  ^  =  4i.  (2) 

Solution.  Multiplying  both  members  of  Eq.  (1)  by  12,  and  of 
Eq.  (2)  by  6,  gives 

4a;- 8- 21  =-3 3^,  (8) 

and  3x  +  4y=27;  (4) 

and,  on  transposing  and  simplifying,  Eq.  (3)  becomes 

4:x+'dy  =  29.  (5) 

Equations  (4)  and  (5)  may  now  be  solved  by  the  method  employed  in 
Ex.  3 ;  and  it  is  easily  verified  that  their  solution,  viz.,  x  =  5  and  y  =  3 
is,  at  the  same  time,  a  solution  of  equations  (1)  and  (2). 

106.  Elimination.  Any  process  of  deducing  from  two  or  more 
simultaneous  equations  other  equations  which  contain  fewer 
unknown  numbers  is  called  elimination.  Such  a  process  elimi- 
nates (i.e.,  gets  rid  of)  one  or  more  of  the  unknown  numbers,  and 
thus  makes  the  finding  of  a  solution  easier. 

That  particular  plan  of  elimination  which  was  followed  in  the 
examples  given  in  §  105  is  known  as  elimination  by  addition  and 
subtraction.  It  is  evident,  moreover,  that  this  method  is  appli- 
cable to  any  pair  of  such  equations.  The  procedure  may  be 
formulated  thus: 

(1)  Unless  each  of  the  ^iven  equations  is  already  in  the 
form  ax  +  by  =  c,  wherein  a,  b,  and  c  are  integers,  reduce 
them  to  this  form. 

(2)  Multiply  these  equations  by  such  numbers  as  will 
mahe  the  coefficient  of  tlxe  letter  to  be  eliminated  the  same 
{in  absolute  value)  in  both  equations. 

(3)  Subtract  or  add  these  last  two  equations  (according 
as  the  terms  to  be  eliminated  have  like  or  unlihe  signs), 
solve  the  resulting  equation  for  the  unhnown  number  which 
it  contains,  and  substitute  that  value  in  any  one  of  the 
earlier  equations  to  find  the  other  unhnown  number. 


168 


ELEMENTARY  ALGEBRA 


[Ch.  XI 


(4)  Verify  that  these  two  numbers  really  satisfy  the  two 
given  equations. 

Note.  If  the  coefficients  which  are  to  be  made  of  equal  absolute  value  are 
prime  to  each  otlier,  then  each  may  be  used  as  a  multiplier  for  the  other  equation ; 
if,  however,  these  coefficients  are  not  prime,  their  least  common  multiple  should 
be  divided  by  each  in  turn,  and  these  quotients  used  as  the  multipliers. 

EXERCISES 

Solve  each  of  the  following  systems  of  equations,  and  check  the  results : 


(15  X 
ox 


15  a;  +  77  2/  =  92, 


3. 


6  ?/  -  5  X  =  18, 
12  x  -  9  ?/  =  0. 

5  a: +  6  2/ =  17, 

6  a: +5  3/ =  16. 


8. 


9. 


^-K2/-2)-K^-3)=0, 
x-\{y-\)-\{x-2)  =  (i. 
^x-ly-m, 


3      4 


^     r5;9  +  3(?  =  68, 
•    12;, 


6. 


7. 


+  5  ^  =  69. 

22  x  -  8  ?/  =  50, 
26  x  +  6  ?/  =  175. 

28  a;  -  23  ^  =  33, 
63  X  -  25  2/  =  199. 

4.<f-i(y-3)  =  5s-3, 
2  y  4-  5  6-  =  69. 


10. 


a: +  3      8-?/^3(a:  +  ?/) 
5  4  8       ■ 

x_y 
4     2 


-2. 


Suggestion.  Eliminate  without  first 
clearing  of  fractions.  When  is  it  advan- 
tageous to  do  this  ? 


11. 


±4-  '1-7 
3  +  3"^' 


6 


61 


12.  What  is  meant  b}'-  saying  that  two  equations  are  simultaneous? 
Consistent?  Independent?  Inconsistent?  Show  the  appropriateness  of 
these  names.     What  is  a  system  of  equations? 

Which  of  these  names  apply  to  the  systems  of  equations  in  the  above 
exercises  ? 


107.  Other  methods  of  elimination.  Besides  the  method  of  elimi- 
nation which  is  explained  in  §  106,  theie  are  several  other 
methods  that  serve  the  same  purpose ;  two  of  these,  which  are 
often  useful,  will  now  be  explained. 


106-107]  SIMULTANEOUS   SIMPLE  EQUATIONS  169 

(i)  Elimination  by  substitution. 

'       ,       ,  ,  r3x-42/  =  7,  (1) 

Ex.1.     Solve  the  system  of  equations    j  o     _ie  xox 

Solution 

FromEq.  (1)  a:  =  ^^^; 

o 

on  substituting  this  expression  for  x,  Eq.  (2)  becomes 

2(^^)  +  32/  =  16;  (3) 

whence  2/  =  2, 

and,  by  substituting  this  value  in  either  of  the  given  equations, 

a:  =  5. 

It  is  easily  verified  that  these  values,  viz.,  x=5  and  y  =  2,  constitute  a 
solution  of  the  given  system  of  equations. 

The  method  of  elimination  which  has  just  now  been  illustrated 
is  known  as  elimination  by  substitution ;  it  is  manifestly  applicable 
to  any  such  system  of  equations  as  the  above. 

The  student  may  solve,  by  this  method,  the  system 
r  3  w  —  4  u  =  19, 
1  5  r«  +  2  y  =  10, 

being  careful  to  check  the  result,  and  then  vrrite  out  a  "rule"  for  applying  this 
method  to  all  such  exercises. 

(ii)  Elimination  by  comparison. 

r  3  a: -4  2/ =7,  (1) 

Ex.  2.    Solve  the  system  of  equations  \ 

^  ^  [2x+3?/  =  16.  (2) 

Solution 

From  Eq.   (1)    x  =  ^—^^,    and    from   Eq.  (2)    x  =  ll^zll.     Now, 
o  2 

since  x  is  to  have  the  same  value  in  each  of  these  equations, 

therefore  7_+4_^  ^  16^-^. 

3  2  •  ^  ^ 

Solving  Eq.  (3)  gives  y  =  2, 

whence,  substituting  this  value  in  either  of  the  given  equations, 

x  =  5. 

It  is  easily  verified  that  these  values,  viz.,  p^  ^  ^  and  y  =  2,  constitute 
a  solution  of  the  given  system  of  equations. 


170 


ELEMENTARY  ALGEBRA 


[Ch.  XI 


The  method  of  elimination  which  has  just  now  been  illustrated 
is  called  elimination  by  comparison ;  it  is  manifestly  applicable  to 
all  such  systems  of  equatibns. 

The  student  may  solve,  by  this  method,  the  system 


8r  +  5i 
12  r- 7; 


and  then  write  out  a  "  rule"  for  applying  this  method  to  all  such  exercises. 


EXERCISES 

Solve  each  of  the  following  systems  of  equations,  using  first  the  method 
of  elimination  by  substitution,  and  then  that  by  comparison,  and  observe 
which  method  is  easier  in  the  different  exercises : 


2  x  +  3  y  =  23, 


34, 
16. 


3.    f  ^  + 

I  5x  +  9?/  =  51. 

(8x-21y  =  SS, 
I  6  a;  +  35  ?/  =  177. 

21y-\-20x  =  165, 


2x  +  y  =  50, 


77 


30  X  =  295. 


8. 


11  <  -  10  r  =  14, 
5t+    7t;  =  41. 


10. 


11. 


12. 


6      7 


2^3        ' 

^  +  •^=.5. 
3     4 

?  +  ^=18, 
o      0 


8       4 
11  r 
12 


13, 
=  12. 


13.  Show  that  elimination  by  comparison  is  merely  a  special  case  of 
elimination  by  substitution. 

14.  State  such  suggestions  as  occur  to  you  for  determining,  by  mere 
inspection,  which  of  the  three  methods  of  elimination  thus  far  considered 
will  be  most  advantageous  to  use  in  any  given  exercise. 


108.  Principles  involved  in  elimination.  Two  systems  of  equa- 
tions (§  104)  are  said  to  be  equivalent  when  every  solution  of  either 
system  is  also  a  solution  of  the  other. 


107-108]  SIMULTANEOUS   SIMPLE  EQUATIONS  171 

The  methods  already  given  (§§  106  and  107)  for  the  solution  of 
a  system  consisting  of  two  independent  equations,  each  containing 
two  unknown  numbers,  consist  in  replacing  a  given  system  of 
equations  by  an  equivalent  system  whose  solution  may  be  more 
easily  obtained.  Those  methods  are  based  upon  the  following 
principles :  * 

(i)  If  any  equation  of  a  system  he  replaced  hy  an  equiva- 
lent equation  (§  95),  the  new  system  thus  formed  will  he 
equivalent  to  the  given  system. 

The  truth  of  this  principle  follows  at  once  from  the  definition  of  equivalent 
systems,  because  if  the  new  equation  has  the  same  solutions,  and  only  those,  as 
the  equation  which  it  replaces,  then  the  new  system  will  have  those  solutions,  and 
only  those,  which  the  given  system  has;  in  other  words,  the  two  systems  are 
equivalent. 

(ii)   //  any  equation  of  a  given  system  he  replaced  hy  the 
equation  formed  hy  adding  ior  suhtracting)  any  otJier  equa- 
tion of  the  system  to  it,  memher  to  memher,  then  the  new 
system  thus  formed  will  he  equivalent  to  the  given  system. 
Proof,    Suppose  the  given  system  of  equations  to  be 

.P  =  0, 

(I)      Q  =  0, 

^  R  =  0, 

wherein  P,  Q,  and  R  represent  polynomials,  — this  is  allowable,  because  if  the 
equations  are  not  in  the  above  form,  they  may  be  brought  to  that  form  by  trans- 
position, which  produces  equivalent  equations  (§95),  and  the  systeins  also  are 
equivalent  [(i)  above];  —  then  it  is  required  to  prove  that  the  above  system  is 
equivalent  to  the  system  p  _|_  q  _  0 

(n)j  Q  =  0, 

^  K  =  0. 

Now  mere  inspection  of  the  two  systems  shows  that  every  set  of  values  of  the 
unknown  numbers  which  satisfies  system  (I),  i.e.,  which  makes  P,  Q,  and  R  each 
separately  0,  also  satisfies  system  (II) ,  and  that  every  solution  of  system  (II)  is 
also  a  solution  of  system  (I) ;  therefore  these  systems  are  equivalent. 

Similarly  in  general. 

(iii)  If  any  equation  of  a  given  system  he  solved  for  one 
of  its  unhnown  numhers,  say  x,  in  terms  of  the  other 
unknown  numhers  which  it  involves,  it  may  he  wi^itten  in 

*  Observe  that  these  principles  apply  to  systems  of  any  number  of  equations  in 
any  number  of  unknown  numbers. 


172  ELEMENTARY  ALGEBRA     '  [Ch.  XI 

the  form  x=  R,  and  this  equation  will  he  equivalent  to  the 
one  which  was  solved  to  obtain  it  (§  95).  If  now  the  expres- 
sion R  he  suhstituted  for  x  in  each  of  tlze  other  equations, 
the  system  of  equations  thus  forined,  together  with  the 
equation  x  =  R,  will  he  equivalent  to  the  given  system. 

To  prove  this  principle,  it  need  only  be  remarked  that  the  only  difference  between 
the  two  systems  of  equations  is  that,  in  the  second  system,  every  x  is  replaced  by 
R,  but,  by  virtue  of  the  equation  x=  R,  every  solution  of  either  system  makes 
the  expression  R  represent  exactly  the  same  number  as  does  x ;  hence  the  two 
systems  are  equivalent. 

109.  Applications  of  the  principles  of  the  preceding  article.     The 

solutions  of  the  exercises  given  in  §§  106  and  107  are  all  based 
upon  one  or  more  of  the  principles  given  in  §  108 ;  this  will  now 
be  illustrated  by  reconsidering  the  solution  of  Ex.  4,  §  105. 


fa;  — 2     . 3  _     ?/ 


(1) 


[j+^f  =  4j.  (2) 

On  multiplying  Eq.  (1)  by  12,  and  Eq.  (2)  by  6,  (I)  becomes 

f4a;-8-21=-37/,  (3) 

l3cc  +  4?/  =  27;  (4) 

and,  on  replacing  Eq.  (3)  by  its  simplified  form,  (II)  becomes 

(III)  (4^ +  3 2/ =  29.  (5) 

i3x  +  4?/  =  27;  (6) 

multiplying  Eq.  (5)  by  4,  and  Eq.  (6)  by  3,  (III)  becomes 

f  16  a; +  12  2/ =  116,  (7) 

I    9x  +  12?/  =  81;  (8) 

replacing  Eq.  (7)  by  the  result  of  subtracting  Eq.  (8)  from  Eq.  (7),  (IV)  becomes 

f7x  =  35,  (9) 

t9x  +  12y  =  81,  (10) 

{x  =  5  (11) 

3  a; +4?/ =  27.  (12) 

But  if  x  =  5,  then  Eq.  (12)  shows  that  y  =  3,  and  hence  a;  =  5  and  ?/  =  3  is  a 
solution  of  system  (VI);  moreover,  (I),  (II),  (III),  and  (IV)  are  equivalent,  by 
§  108  (i) ;  (IV)  is  equivalent  to  (V),  by  §  108  (ii) ;  and  (V)  is  equivalent  to  (VI), 
by  §  108  (i) ;  hence  (I)  and  (VI)  are  equivalent,  and  therefore  x  =  5  and  y  =  '3 
is  a  solution  of  (I)  also. 


108-109]  SIMULTANEOUS   SIMPLE  EQUATIONS 


173 


EXERCISES 


Solve  each  of  the  following  S3-stems  of  equations.     In  the  solution  of 
the  first  ten,  give  detailed  explanations  like  those  given  in  §  109  : 


^r7a;  +  47/  =  l, 
l9x  +  4?/  =  3. 

2  |3a:  +  5?/=19, 
I  5  x  -  4  ?/  =  7. 

3  ja:-ll?/  =  l, 
llly-9x  =  99. 


4.^ 


21  y 


I  G  M  +  14  y  =  -  26. 

'  I  51  a;  +  25  y  =  101. 

g   (Sdx-loy  =  93, 
65  X  +  17  y  =  113. 


7.^ 


r  19  5  +  85  f  =  350, 


1 17  6- +  119^  =  442. 


i2r 


11  MJ  =  0, 

17  w  =  139. 


9. 


10. 


r    3  a:-  ll?/  =  0, 
I  19  x  -  19  y  =  8. 


2      3       ' 
X  _2y 
4       3 


3. 


11. 


3      6      2 
•r  _  3j  ^  _  1 
I  5      10  2 


12.  ^ 


13. 


+  3  7/  +  14  =  0, 


-  +  5?/  +  4  =  0. 
5 


+  5  ;2  =  -  4, 


||+5.= 


4. 


14.-^ 


15.  i 


16. 


17. 


18. 


19.^ 


(  x±2_ 
3 


+  4?/ 


7/ +11      3;  +  l^.. 
11  2 


2r  +  3^      ^+<^  =  o 
5  7"' 


2 


_5_^  ^  r  +  7 


3  4 

m-2      n+2 


ll-2n 


0. 


li  -2      ^  +  5 

3  2 

2^-7      13  -  A- 


0, 


12  = 


6 

+  32 


10. 


?  +  ^ 


^^y 


25. 


.2  ?/  +  .5  _  .49  a:  -  .7 

1.5  ~       4.2 

.5  a:-  .2^41      1.5  y-  11 

1.6  16  8 


174 


ELEMENTARY  ALGEBRA 


[Cii.  XI 


20. 


21. 


V  +  i(3v  -  w-  1)  = 

(^f.n)-(4x. 

3x - 5y      2x -8y 

i  +  K^- 

«  +  24). 
-  9     31 

1), 

5 
=  6' 

3 

12 

12 

110.  Simultaneous  fractional  equations.  By  first  clearing  the 
given  equations  of  fractions  (§§  98  and  99),  the  foregoing 
methods  become  applicable  to  the  solution  of  fractional  equa- 
tions,—  the  following  examples  will  illustrate  this. 

3 


Ex.  1.     Given  the  system  of  equations  . 


^1  +  1=- 

X      y      X 


y    xy } 


to  find  X  and 


Solution 

On  multiplying  each  of  these  equations  by  xy,  they  become,  respectively, 

y  +  X=z3y, 

and  Q  y  —  X  =  1. 

The  solution  of  these  integral  equations  is  (§  106)  x  =  \  and  y  =  \', 

and  it  is  easily  verified  that  these  numbers  constitute  a  solution  of  the 

given  system  of  fractional  equations  also. 

r       1        ^  4^        If) 

^ 3  ?y      X      x(x 3  y/^ 

Ex.  2.     Given  the  system  of  equations  ]  - ;  to 

find  X  and  y.  £_1_7/  —  0 

o 
Solution 

On  multiplying  these  equations  by  x{x  -Sy)  and  3,  respectively,  they 

become  x  +  4:  (x  -  S  y)  =  16, 

and  x-S-Sy  =  0. 

By  §  106  the  solution  of  this  system  of  integral  equations  is  a:  =  4  and 
y  =  I,  and  these  two  numbers  prove  also  to  be  a  solution  of  the  given 
system  of  fractional  equations. 


Ex.  3.     Given   the    system   of    equations 
solution. 


1  +  1  =  3 

X      y 

?-5  =  i 


;   to  find    its 


109-110]  SIMULTANEOUS   SIMPLE  EQUATIONS  175 

Solution 

On  multiplying  each  of  these  equations  by  xy,  they  become,  respectively, 

y  +  x  =  Sxy, 

and  2y  —  3x  =  xy; 

if  the  first  of  these  be  subtracted  from  three  times  the  second,  the  result 
will  be 

5y-  10x  =  0, 

i.e.,  y  =  2x. 

On  substituting  this  value  of  y  in  the  first  of  the  given  equations,  it 

becomes  £  4.  J_  =  3, 

x     2  X 

whence,  multiplying  by  2  ar,     2  +  1  =  6  a;, 

i.e.,  X  =  h 

and  therefore,  since  y  =  2  x,  y  =  ^^ 

It  is,  moreover,  easily  verified  that  x  =  I  and  y  =  1  constitutes  a  solution 
of  the  given  fractional  equations. 

Note.     Solve  Ex.  3  by  eliminating  before  clearing  of  fractions  (e.g.,  subtract 
the  second  equation  from  twice  the  first),  and  compare  the  two  methods. 


Ex.  4.     Given  the  system  of  equations    . 
X  and  y. 

Solution 


^2 


x-1      y-2 


■ ;  to  find 


On  multiplying  the  first  of  these  equations  by  x  and  the  second  by 
(x  —  l)(y  —  2),  and  simplifying,  they  become,  respectively, 

xy  -4:x  +  2  =  0, 

and  2xy~ox-3y  +  7  =  0. 

To  eliminate  the  term  containing  xy  subtract  twice  the  first  of  these 
integral  equations  from  the  second ;  the  result  is 

3x-dy  +  d  =  0, 

i.e.,  y  =  x  +  1. 


176 


ELEMENTARY  ALGEBRA 


[Cii.  XI 


On  substituting  this  value  of  y  in  the  first  of  the  integral  equations, 

it  becomes  /     ,   i  \      i      ,  o      n 

X  (x  +  1)  -  4  a;  +  2  =  0, 

i.e.,  x^-Sx  +  2  =  0, 

i.e.,  (x-l){x-2)=0, 

whence  a:  =  1  or  a;  =  2 ; 

and  since  y  =  x  +  1,  therefore  the  corresponding  values  of  y  are 

y  =  2  and  ^  =  3. 

While  each  of  these  pairs  of  corresponding  values,  viz.,  x  =  1,  y  =  2 
and  x  =  2,  ?/  =  3,  is  a  solution  of  the  system  of  integral  equations  obtained 
by  clearing  the  given  system  of  fractions,  yet  it  is  easily  verified  that  the 
second  pair  is  a  solution  of  the  given  system  of  fractional  equations  and 
that  the  first  pair  is  not  a  solution  of  this  system. 

Observe  that  extraneous  solutions,  here  as  in  §  99,  reduce  one  or  more 
of  the  denominators  to  zero. 

EXERCISES 

Solve  the  following  systems  of  equations,  and  check  the  results; 
eliminate  before  clearing  of  fractions  when  that  is  possible,  as  in 
Exs.  8-11,  13,  etc. : 


3  a:  +  2  V  +  6 


4a;- 

2y 

-  ^f 

3  -  7  ?/ 
2x  +  l 

=  2. 

15-f  .V 

-2x 

-  5 

4  a:  —  5 

y-2 

3a;-2 

y  +  ^. 

16 

X  — 

y 

3 

8 

15 

6. 


7.    -jSx+lOy      '3  X  —  4:  y 


x  =  7. 


X      y 


=  1. 


9. 


10. 


11. 


12. 


5  +  «: 

X     y 

6  .  5 


20, 
10. 


--5=5, 


-2  =  7. 


1-1  +  3. 

2  V     10 

A+i=23. 

2v     w 

1  2 


a: -2      3 

i  3  a:  +  ?/  = 


*  Show  that  the  equations  in  Ex.  12  are  inconsistent. 


110-111]  SIMULTANEOUS   SIMPLE  EQUATIONS 


177 


13. 


14. 


3 2       ^13 

2x-o     3ij  +  2       5 ' 

-1 ^  =  8. 

2a:-  5     2  +  3  </ 

3  '^       ^  2, 

4:U  +  V        2  U  —   0 

3       ^       4 


23 


2  u  —  V     V  +  4  u 


15.     ^ 


x      q_5?/-f2x     x  —  3 


2 

2y-3y 

y  +  \ 


+  y  =  l2 


19. 


x-20 


16. 


17. 


4x  + 


2       16X  +  19 


17 -3x 


50_    y-1    ^s       147-24.y, 
|(a;-2)        ^^         3 


\x-2     3?/  + 
5^/  + 


2x-7 


f  3 


9  _ 


18.    ^ 


1 


7  -2y  

2      *  [u-l'  V  ^^' 

2  ?/  -  X  _  2  a:  -  59 
23  -  x  ~       2       ' 


20. 


"^      x-lS  3 

|(2x  +  3)+l|^^=31  +  ^^, 

8y  +  7     6x-3?/^  .      4?/-9 
[      10         2(2^-4)  5 


111.  Literal  equations.  Literal  equations  of  the  first  degree,  and 
involving  but  one  unknown  number,  have  already  been  discussed 
(§  97) ;  the  present  article  will  be  devoted  to  the  consideration  of 
a  pair  of  simultaneous,  independent,  literal  equations  of  the  first 
degree,  each  involving  two  unknown  numbers. 

Since,  by  transposing  and  collecting  terms,  every  first  degree 
equation  in  two  unknown  numbers  may  be  reduced  to  an  equiva- 
lent equation  of  the  form  ax  -\-hy  =  c,  wherein  a,  6,  and  c  represent 
known  numbers,  therefore  the  two  given  equations  will  be  assumed 
as  already  in  that  form.     It  is  then  proposed  to  solve  the  system 

of  equations 

I  a^x  +  h^y  =  Ci,  (1) 

1  a.x  4-  h^y  =  Cg.  (2) 


*  Compare  Ex.  4  above,  and  use  §  66  (iv)  if  necessary. 


178  ELEMENTARY  ALGEBRA  [Ch.  XI 

To  eliminate  y  multiply  Eq.  (1)  by  62;  and  Eq.  (2)  by  h^  and 
then  subtract ;  this  gives 

{oLyb^  —  a^-^x  =  62C1  —  61C2,  (3) 

whence  x  =  -^ ^^-  (4) 

Similarly,  by  eliminating  x, 

(ai&2  -  ^2^)2/ =  aiCs  -  a2Ci,   «  (p) 

whence  0,0, -a,c. 

It  is  also  easily  verified  (by  substitution)  that  these  expressions 
for  X  and  y  satisfy  the  given  system  of  equations.  Hence  the 
given  system  of  equations  has  at  least  one  solution,  provided  only 
that  a^2  —  «'2^i  ^  0.* 

Moreover,  by  §  108,  the  system  consisting  of  Eqs.  (4)  and  (6)  is 
equivalent  to  the  given  system ;  but,  for  any  given  set  of  values 
of  the  coeJBficients,  Eqs.  (4)  and  (6)  have  manifestly  but  one  solu- 
tion, and  hence  the  given  system  has  hut  one  solution. 

Hence,  any  system  consisting  of  two  independent  and  con- 
sistent first  degree  equations,  involving  two  unknown  num- 
bers, has  one  solution,  and  but  one. 

Note.  It  may  also  be  stated  here  that  three  or  more  independent  equations  of 
the  first  degree,  involving  only  two  unknown  numbers,  can  not  all  be  satisfied  by 
the  same  values  of  the  unknown  numbers. 

For,  if  the  solution  of  the  first  two  of  these  equations  is  a  solution  of  the  third 
equation  also,  then 

\a1b2-a2b1J  \aib2-a.2bi) 

i.e.,  in  this  case,  there  is  a  definite  relation  (equation)  connecting  the  coefficients 
of  the  given  equations,  and  these  equations  are,  therefore,  not  independent. 
Similarly  in  general. 

*  If  0162  —  02^^1  =  0,  then  X  (=  h£izz]h£fi.\  is  infinite,  unless  it  happens  that 
\     a1b.2~a.2b1 1 

62C1—  61C2  is  also  0,  in  which  case  ^  =  2l=:£i,  and  the  given  equations  are  not 

02      62      C2 
independent,  for  either  of  them  may  then  be  obtained  by  multiplying  the  other  by 
a  suitable  factor  ;    i.e.,  in  this  case  there  is  really  only  one  equation  and  the 
number  of  solutions  is  infinite  (§  101) . 


Ill] 


SIMULTANEOUS   SIMPLE  EQUATIONS 


179 


EXERCISES 

Solve  the  following  systems  of  equations,  and  check  the  results ;  elimi- 
nate without  clearing  of  fractions  when  possible : 


1. 


2.    -i 


4. 


5. 


'  ax  +  by 

=  m, 

[  bx  4-  ay 

=  n. 

f  --  y 

=  a  — 

b, 

[  ax  +  by 

=  a^- 

-62. 

("  +  »  = 

X      y 

1 
c 

c      a  _ 
X      y~ 

1 
b 

ax      by 

1 

bx      cy 

1 
a2' 

«      + 
a^  X 

b 
b-\-y 

c 

c  +  1' 

h 
a-^x 

c 

b  +  y 

a 

a+  1 

6. 


X  +  y  _  ^  -y  ^  0, 

=  0. 


a  b 

X  —  a  _  y  —  b 


2a'^-  ax=2b'^  +  by, 


7.    i      y 


b     a  +  b 


a  +  b 
ab 


8.    i 


{  {a-^b)x-\-{a^c)y  =  a^b, 


[  {a-\-c)x-\-{a-\-b)y  =  a  +  c. 

y  +  l~  a-b  +  V 
X-  v  =  2b. 


10. 


hx  +  ky  =  4:  h% 


+ 


h 


X  —  k      y  —  h,      k(y  —  h) 

11.  Under  what  circumstances  has  Ex.  1  above  no  finite  solution? 
Answer  this  question  with  regard  to  Ex.  2  also ;  and  with  regard  to  Ex.  7. 

12.  What  relation  among  the  coefficients  is  needed  in  order  that  Ex.  1 
shall  have  more  than  one  solution?  If  this  relation  exists,  how  many- 
solutions  has  this  system  of  equations  ? 

13.  What  relation  among  the  coefficients  is  required  in  order  that  the 
three  equations  ax  +  by  =  c,  bx  +  cy  =  a,  and  ex  +  ay  =  b  may  have  one 
solution  in  common  ? 

PROBLEMS 

1.    Find  two  numbers  whose  difference  is  -^^  of  their  sum,  and  such 
that  5  times  the  smaller  minus  4  times  the  larger  is  39. 


Let 
and 


Solution 
z  =  the  larger  number, 
y  =*  the  smaller  number. 


180  ELEMENTARY  ALGEBRA  [Ch.  XI 

Then,  by  the  conditions  of  the  problem, 

•^         35   ' 
and  5 1/  —  4  a;  =  39. 

Solving  these  equations,  we  obtain 

a;  =  54  and  y  =  5l; 
and  these  numbers,  which  constitute  a  solution  of  the  equations  of  the  problem, 
also  satisfy  ail  the  conditions  of  the  problem  itself,  and  are,  therefore,  the  num- 
bers sought. 

2.  Find  two  numbers  such  that  3  times  the  greater  exceeds  twice  the 
less  by  29,  and  twice  the  greater  exceeds  3  times  the  less  by  1. 

3.  A  lady  purchased  20  yds.  of  one  kind  of  goods,  and  50  yds.  of 
another,  for  |3();  she  could  have  purchased  30  yds.  of  the  first  kind,  and 
20  of  the  second,  for  |23.     What  was  the  price  of  each? 

4.  If  A's  money  were  increased  by  $4000,  he  would  have  twice  as 
much  as  B.  If  B's  money  were  increased  by  $5500,  he  would  have 
3  times  as  much  as  A.     How  much  money  has  each? 

5.  One  eleventh  of  A's  age  is  greater  by  2  years  than  ^  of  B's,  and 
twice  B's  age  equals  what  A's  age  was  13  years  ago.  Find  the  ages  of 
each. 

6.  If  45  bushels  of  wheat  and  37  bushels  of  rye  together  cost  $62.70, 
and  37  bushels  of  wheat  and  25  bushels  of  rye,  at  the  same  prices,  cost 
$48.30,  what  is  the  price  of  each  per  bushel? 

7.  A  pound  of  tea  and  6  lb.  of  sugar  together  cost  72  cents ;  if  sugar 
were  to  advance  50%,  and  tea  10%,  then  2  lb.  of  tea  and  12  lb.  of  sugar 
would  cost  $  1.68.     Find  the  present  price  of  tea,  and  also  of  sugar. 

8.  A  man  having  $45  to  distribute  among  a  group  of  children,  finds 
that  he  lacks  $1  of  being  able  to  give  $3  to  each  girl  and  $  1  to  each 
boy,  but  that  he  has  just  enough  to  give  $2.50  to  each  girl  and  $1.50  to 
each  boy.     How  many  boys  and  how  many  girls  are  there  in  this  group? 

9.  John  said  to  James,  "Give  me  8  cents  and  I  shall  have  as  much 
as  you  have  left."  James  said  to  John,  "  Give  me  16  cents  and  I  shall 
have  4  times  as  much  as  you  have  left."     How  much  money  had  each? 

10.  A  boy  bought  some  oranges  at  the  rate  of  3  for  5  cents,  and  another 
kind  at  4  for  5  cents,  and  paid  for  the  whole  $4.60.  He  afterwards  sold 
them  all  at  2  cents  apiece,  clearing  thereby  $1.54.  How  many  of  each 
kind  did  he  buy? 

11.  A  fishing  rod  consists  of  two  parts ;  the  length  of  the  upper  part 
is  ^  that  of  the  lower  part;  the  sum  of  9  times  the  length  of  the  upper 
part  and  13  times  the  length  of  the  lower  part  exceeds  11  times  the  length 
of  the  whole  rod  by  36  inches.     P'ind  the  length  of  the  rod. 


Ill]  SIMULTANEOUS   SIMPLE  EQUATIONS  181 

12.  If  a  certain  rectangular  floor  were  2  ft.  broader  and  3  ft.  longer, 
its  area  would  be  increased  by  64  sq.  ft.,  but  if  it  were  3  ft.  broader  and 
2  ft.  longer,  its  area  would  be  68  sq.  ft.  greater  than  it  now  is.  Find  its 
length  and  breadth. 

13.  Three  rectangles  are  equal  in  area;  the  second  is  6  meters  longer 
and  4  meters  narrower  than  the  first,  and  the  third  is  2  meters  longer  and 
1  meter  narrower  than  the  second.    What  are  the  dimensions  of  each? 

14.  The  sum  of  the  ages  of  a  father  and  son  will  be  doubled  in  25 
years,  and  20  years  hence  the  difference  of  their  ages  will  just  equal  ^  of 
their  sum  at  that  time.     What  is  the  present  age  of  each  ? 

15.  If  1  be  added  to  each  term  of  a  certain  fraction,  its  value  will  be  |; 
but  if  1  be  subtracted  from  each  of  its  terms,  its  value  will  be  ^.  What 
is  the  fraction? 

16.  The  sum  of  the  digits  of  a  two-digit  number  is  12,  and  if  its  digits 
be  interchanged,  the  number  thus  formed  will  lack  12  of  being  the  double 
of  what  it  now  is.     What  is  the  number? 

17.  If  a  certain  two-digit  number  is  divided  by  the  sum  of  its  digits, 
the  quotient  is  8,  and  when  the  tens'  digit  is  diminished  by  3  times  the 
units'  digit,  the  remainder  is  1.     What  is  the  number? 

18.  The  tickets  of  admission  to  an  entertainment  were  50  cents  for 
adults  and  35  cents  for  children.  If  the  proceeds  from  the  sale  of  100 
tickets  was  $39.50,  how  many  tickets  of  each  kind  were  sold? 

Solve  this  problem  also  by  using  but  one  letter  to  represent  an 
unknown  number. 

19.  A  capitalist  invested  $4000,  part  of  it  at  5%  and  the  balance  at  4%, 
and  found  that  his  annual  income  from  this  investment  was  $175.  How 
much  was  invested  at  5%,  and  how  much  at  4%? 

Can  this  problem  be  solved  without  using  two  letters  to  represent 
unknown  numbers?     How? 

20.  A  boat  crew  can  row  4  miles  downstream  and  back  again  in 
U  hours,  or  6  miles  downstream  and  halfway  back  in  the  same  time. 
What  is  the  rate  of  rowing  in  still  water,  and  what  is  the  rate  of  the 
current? 

21.  A  capitalist  invested  1^4,  part  at  p%  and  the  balance  at  q%,  and 
found  that  his  annual  income  from  this  investment  was  1 5.  How  much 
was  invested  at  />%? 

Show  that  this  problem  includes  Prob.  19  as  a  special  case  —  it  is  the 
generalization  of  Prob.  19  (cf.  §  100). 


182  ELEMESTARY  ALGEBRA  [Ch.  XI 

22.  Generalize  Prob.  14.  Find  the  solution  of  the  generalized  problem, 
and  then  show  that  the  answer  to  the  particular  problem  (14)  may  be 
found  by  merely  substituting  in  the  answer  to  the  generalized  problem. 

23.  Generalize  Prob.  20,  solve,  etc.,  as  in  Prob.  22. 

24.  A  man  rows  15  miles  downstream  and  back  in  11  hours.  If  he 
can  row  8  miles  down^^tream  in  the  same  time  as  it  takes  him  to  row 

3  miles  upstream,  what  is  his  rate  of  rowing  in  still  water?  and  what  is 
the  velocity  of  the  current? 

25.  Divide  the  number  N  into  two  such  parts  that  —  of  the  first 

1  *" 

part,  plus  -  of  the  second,  shall  exceed  the  first  part  by  M. 
n 

Specialize  this  problem,  and  find  the  solution  of  the  special  problem 

by  substituting  in  the  general  solution. 

26.  Three  cities.  A,  B,  and  C,  are  situated  at  the  vertices  of  a  triangle; 
the  distance  from  A  to  C  by  way  of  B  is  50  miles,  from  A  to  B  by  way 
of  C  is  70  miles,  and  from  B  to  C  by  way  of  A  is  60  miles.  How  far 
apart  are  these  cities? 

Solve  this  problem  by  first  generalizing  it,  and  then  substituting  the 
particular  numbers  50,  70,  and  60  in  the  general  solution. 

27.  Two  boats  which  are  d  miles  apart  will  meet  in  a  hours  if  they 
sail  toward  each  other,  and  the  second  will  overtake  the  first  in  b  hours 
if  they  sail  in  the  same  direction.  Find  the  respective  rates  at  which 
these  boats  sail.  Also  discuss  fully  your  solution,  i^^  interpret  the  results 
when  the  rate  of  the  second  boat  is  greater  than,  equal  to,  and  less  than, 
the  rate  of  the  first  —  compare  Prob.  3  of  §  100. 

28.  Two  men,  A  and  B,  had  a  certain  distance  to  row  and  alternated 
in  the  work ;  A  rowed  at  a  rate  sufficient  to  cover  the  entire  distance  in 
10  hours,  while  B*s  rate  would  require  14.  If  the  journey  was  completed 
in  12  boursL  how  many  hours  did  each  row? 

29.  A  mine  which  is  to  be  emptied  of  water  has  two  pumps  which 
together  can  discharge  1250  gallons  an  hour.  The  larger  pump  can  do 
the  work  alone  in  5  hours,  but  with  the  help  of  the  smaller  pump  only 

4  hours  are  needed.   How  many  gallons  an  hour  does  each  pump  discharge  ? 

Solve  this  problem  by  first  generalizing  it,  as  in  Prob.  26  above. 

30.  Two  trains  are  scheduled  to  leave  the  cities  A  and  B,  m  miles 
apart,  at  the  same  time,  and  to  meet  in  h  hours;  but,  the  train  leaving  A 
being  a  hours  late  in  starting,  they  met  k  hours  later  than  the  scheduled 
time.    What  is  the  rate  at  which  each  train  runs  ? 

From  the  solution  of  this  problem  find,  by  substitution,  the  solution 
of  the  special  problem  in  which  m  =  800,  ft  =  10,  a  =  If,  and  k  =  ^. 


111-112]  SIMULTANEOUS   SIMPLE  EQUATIONS  183 

31.  Two  boys,  A  and  B,  run  a  race  of  400  yards,  A  giving  B  a  start  of 
20  seconds  and  winning  by  50  yards.  On  running  this  race  again,  A, 
giving  B  a  start  of  125  yards,  wins  by  5  seconds.  What  is  the  speed  of 
each?     Generalize  this  problem. 

32.  A  and  B  working  together  can  build  a  wall  in  5^  days ;  finding 
it  impossible  to  work  at  the  same  time,  A  works  5  days,  and  later  B  takes 
up  the  work,  finishing  it  in  6  days.  In  how  many  days  could  each  have 
built  this  wall  alone?     Generalize  this  problem. 

33.  A  railway  train,  after  running  1  hour  and  36  minutes,  was  detained 
40  minutes  by  an  accident,  after  which  it  proceeded  at  |  of  its  former 
rate,  and  reached  its  destination  16  minutes  late.  Under  the  same  cir- 
cumstances, had  the  accident  occurred  10  miles  farther  on,  the  train 
would  have  arrived  20  minutes  late.  At  what  rate  did  the  train  move 
before  the  accident,  and  what  was  the  entire  distance  traveled? 

II.  THREE  OR  MORE  UNKNOWN  NUMBERS 

112.  Equations  containing  more  than  two  unknown  numbers.  It 
is  easy  to  see  that  the  methods  employed  in  §  105  for  solving  a 
system  of  two  simultaneous  integral  equations,  each  containing 
two  unknown  numbers,  may  also  be  employed  for  solving  a  system 
of  three  or  more  such  equations  involving  as  many  unknown  num- 
bers as  there  are  independent  equations.     (Cf.  Exs.  1  and  2  below.) 

(     x-i-Si/-     z  =  o,  (1) 

Ex.  1.    Given  |  3  x  +  6  ?/  +  2  z  =  .3,  (2) 

to  find  the  solution  of  this  system  of  equations. 

Solution.  Adding  2  times  Eq.  (1)  to  Eq.  (2),  member  to  member, 
g^^^s  5x+V2y=lS,  (4) 

and  subtracting  Eq.  (3)  from  3  times  Eq.  (1)  gives 

x+12t/=9.  (5) 

Now  subtracting  Eq.  (5)  from  Eq.  (4)  gives 

4  a:  =  4, 
whence  x  =  1.  (6) 

On  substituting  this  value  of  x,  Eq.  (5)  becomes 

H-12y  =  9, 
whence  v  =  i\  C7\ 


184   .  ELEMENTARY  ALGEBRA  [Ch.  XI 

and  substituting  these  values  of  x  and  y  in  Eq.  (1)  gives 

whence  z  =  —  2.  (8) 

That  these  numbers,  viz.,  x  =  1,  y  =  ^,  and  z  =  —  2,  really  constitute  a 
solution  of  the  given  system  of  equations  is  easily  verified  by  substituting 
them  for  x,  y,  and  z  in  these  equations. 

Note.  It  should  be  carefully  observed  that,  by  principles  (i)  and  (ii)  of  §  108, 
Eq.  (2)  of  the  given  system  of  equations  may  be  replaced  by  Eq.  (4),  —which  is 
derived  from  Eq.  (1)  and  (2),  —  and  the  new  system  thus  formed  will  be  equivalent 
to  the  given  system,  i.e.,  the  system  of  Eqs.  "(1),  (3),  and  (4)  is  equivalent  to  the 
system  of  Eqs.  (1),  (2),  and  (3). 

So  too  Eq.  (3)  may  be  replaced  by  Eq.  (6),  making  the  system  formed  of  Eqs.  (1), 
(4),  and  (0)  equivalent  to  the  given  system;  and  this  last  system,  being  readily 
solved,  furnishes  a  solution  of  the  given  system. 

The  foregoing  is  another  illustration  of  the  fact  to  which  attention  has  already 
been  called  (§  108),  viz.,  that  solving  a  system  of  simultaneous  equations  is 
accomplished  by  fix-st  replacing  the  given  system  by  an  equivalent  system  whose 
solution  is  more  easily  obtained. 

(2x-^y-2z  =  -l,  (1) 

Ex.  2.   Given      '  |  3  a:  +  z  =  6,  (2) 

i  a:  +  2/  +  2  =  3  ;  (3) 

to  find  the  solution  of  these  equations. 

Solution.  Since  the  second  of  these  equations  is  already  free  from 
the  unknown  number  y^  therefore  it  is  best  to  combine  Eqs.  (1)  and  (3) 
so  as  to  eliminate  y,  and  thus  obtain  another  equation  involving  only  x 
and  2.     Adding  Eq.  (1)  to  3  times  Eq.  (3)  gives 

5  a:  +  2  =  8,  (4) 

and  subtracting  Eq.  (2)  from  Eq.  (4)  gives 

2  a:  =2, 
whence  x  =  \.  (5) 

Substituting  this  value  of  x  in  Eq.  (2)  gives 

2  =  3; 
and  substituting  these  two  values  in  Eq.  (3)  gives 

2/  =  -l. 
Moreover,  it  is  easily  verified  that  a;  =  1,  ^  =  —  1,  and  2=3  constitute 
a  solution  of  the  given  equations. 

Ex.  3.  Show  that  Eqs.  (2),  (3),  and  (5),  in  Ex.  2,  form  a  system  which 
is  equivalent  to  the  given  system. 


112-113]  SIMULTANEOUS   SIMPLE  EQUATIONS  185 

113.  Formulation  of  the  method  of  procedure  of  §  112.  The  proc- 
ess of  finding  a  solution  of  three  independent  integral  equations 
of  the  first  degree  and  containing  three  unknown  numbers,  which 
is  illustrated  in  §  112,  may  be  stated  thus : 

Combine  any  two  of  the  three  given  equations  in  such  a 
way  as  to  eliminate  some  one  of  the  unknown  numbers, 
thus  deriving  from  them  an  equation  containing  but  two 
unhnown  nujnbers;  then  combine  the  remaining  equation 
of  the  given  system  with  either  one  of  the  other  two  in  such 
a  way  as  to  eliminate  the  same  unhnoiun  number  as  before, 
thus  deriving  another  equation  which  contains  the  same 
two  unknown  nuinbers  as  does  the  first  derived  equation; 
next  combine  these  two  derived  equations  so  as  to  elimAnate 
one  of  the  unknown  numbers,  thus  deriving  another  equa- 
tion which  contains  but  one  unknown  number;  from  this 
last  equation  the  value  of  the  unknoiun  number  ivhich  it 
contains  can  be  found,  and  then,  by  successively  substituting 
in  earlier  equations,  the  values  of  the  other  two  unknown 
numbers  can  be  found. 

Similarly  for  the  solution  of  a  system  of  n  independent  integral 
equations  of  the  first  degree  and  containing  n  unknown  numbers. 
When  n  is  greater  than  3  the  eliminating  should  be  done  very 
systematically,  since  otherwise  the  derived  equation  may  not  be 
independent ;  the  procedure  may  be  stated  thus  : 

So  combine  some  one  of  the  given  equations  {the  first,  for 
example)  with  each  of  the  others,  as  to  eliminate  tJie  same 
unknoivn  number  in  each  case,  thus  forming  ivhat  may 
be  called  a  first  derived  system  of  n  —  1  equations,  which 
will  be  independent,  integral,  and  of  the  first  degree,  and 
which  will  contain  n  —  1  unknown  numbers ;  by  proceeding 
with  the  first  derived  system  just  as  with  the  given  sys- 
tem, a  second  derived  sy stein  containing  n  —  2  equations 
involving  n  —  2  unknown  numbers  is  obtained;  by  continu- 
ing this  process,  there  is  finally  obtained  a  single  equation 
with  but  one  unknown  number;  from  this  equation  the 
value  of  that  unknown  number  is  found,  and  then,  by 


186 


ELEMENTARY  ALGEBRA 


[Ch.  XI 


successive  substitutions  in  earlier  equations,  the  values  of 
all  the  other  unknown  numbers  are  found. 

Note.  It  may  be  remarked  that  any  one  of  the  give)i  equations,  together  with 
then  — 1  equations  of  the  first  derived  system,  constitute  a  system  which  is 
equivalent  to  the  given  system  ;  also  that  any  one  of  the  given  system,  together 
with  any  one  of  the  first  derived  system,  and  the  n  — 2  equations  of  the  second 
derived  system,  are  equivalent  to  the  given  system,  and  so  on;  finally,  that  the 
system  composed  of  any  one  of  the  given  equations,  any  one  of  the  first  derived 
system,  any  one  of  the  second  derived  system,  and  so  on  including  the  single 
equation  of  the  last  derived  system,  is  equivalent  to  the  given  system. 


EXERCISES 


Solve  each  of  the  following  systems  of  equations : 


1. 


2. 


3. 


4. 


2:c  +  3?/  +  4z 
3a:  +  5?/+6^ 


20, 
26, 
31. 


4  a;  —  y  —  z  =  5, 

3a;-4?/+16  =  6z, 

lx  +  ^y-2z  =  lQ, 
2a;  +  5y  +  3^  =  39, 

^  X  —     y  -\-  5  z  =  31. 

5x-Qy  +  4:Z=l5, 
7  X  +  4:y-  3z=19, 
2x  +     y+Qz  =  i6. 

2  X  +  4  ?/  +  5  2  =  19, 

Sx-Sy  +  bz  =  2S. 

5x  +  6y  -12z  =  6, 
2x-2y  -    62=- 1, 
4x-52/+    32  =  7^. 

y  +  z  -8Q  =  72  -  5x, 
9'^-ix-\y  =  iy-2z, 
lx  +  ly  +  lz=5S. 


10. 


12. 


^x  +  ly=12-iz, 
iy+  Xz  =  8+lx, 
ix  +  iz  =  10. 

|'2a;-5?/+19  =  0, 
rdy-4:z-{-  7  =  0, 
\2z-5x-    2  =  0. 


0  4 


3, 


4      o 


5. 


'1  +  ^  =  6, 

X        V 


11.  \ 


1    1 

-  +  - 


10, 


1      1 

-+  -  = 

Z        X 


3      2      1, 

^  +  -  +  -  =  1, 
X      y      z 


V      z 


=  1. 


113j 


SIMULTANEOUS   SIMPLE  EQUATIONS 


187 


13. 


14. 


(  X  +  y  -z  =  a, 
{x-y  =  2b, 
[x  +  z  =  Sa-^h, 

X  -^  y     a 

yz     ^1 

y  +  z     6' 

xz         1 
c 

If 


18. 


19.  i 


x± 
xy 


[x+  z 
Suggestion 
=  a,  i.e., 


— h-  =  o 

y     X 


xy 


then 


15. 


16 


2  y  +  3  a;  +     y  -     2  =  0, 

3  3/  ^  2  X  +     2  -  4  <;  =  21, 
22-3y-     ?/+     a:  =6, 

r  +  4  a:  +  2  ?/  -  3  2  =  12. 

u  +  a:  +  //  =  15, 

X  +  ?/  +  2  =  18, 

y  +  _y  +  2  =  17, 

L  I'  4-  X  +  2  =  16. 

Suggestion.      Adding    these    equa- 
tions and  dividing  the  sum  by  3  gives 

v-\-x-\-y  +  z  =  22,. 


{  y  +  z-Zx  =  2a, 
x  +  z-3y  =  2b, 
X  +  y  —  S  z  =  2  c, 

I2x-h2y+v  =  0. 

(Zu  +  ov-2x  +  ^z  =  2, 
2u  +  4x-^y-z  =  Z, 

u  -\-  V  +  z  =  2, 
6y  +  iv  +  u  =  2, 
5z  +  4:x-7v  =  0. 


«        1         1        o 

-  +  -+-  =  2, 
X     y      z 

20.   J  1      h      2      ^ 

X     y      z 

yz  +  xz  +  cxy  =  3  xyz. 

Suggestion.  Carefully  compare  the 
last  equation  with  either  of  the  other 
two. 


21. 


22. 


abxjjz  4-  cxy  —  ayz  =  bxz, 
bcxyz  +  ayz  —  bxz  =  cxy, 
acxyz  +  bxz  —  cxy  =  ayz. 


5xy  -\-  Q(x+y)=  0, 

5yz-2(y  +  z)=0, 

Uxz  -  3(a:+  z)=0. 


17 


(  y  +  z  +  V  —  X  =  22, 

1z  +  V  +  X  -  y  =  18, 
V  +  X  -\-  y  —  z  =  li, 
X  +  y  -j-  z  —  V  =  10. 

23.  From  the  considerations  presented  in  §  113,  prove  that  a  system 
consisting  of  n  independent  and  consistent  equations  of  the  first  degree, 
and  containing  n  unknown  numbers,  has  one  and  only  one  solution.  (Cf. 
also  §  111.) 

24.  If  there  are  more  unknown  numbers  than  independent  equations 
in  any  given  system,  how  many  solutions  has  that  system?  Why  ?  (Such 
a  system  is  usually  called  an  indeterminate  system.) 

25.  If  there  are  more  consistent  equations  than  unknown  numbers  in 
a  system,  prove  that  these  equations  can  not  all  be  independent.  (Cf. 
§  111,  note.) 


188  ELEMENTARY  ALGEBRA  [Ch.  XI 

26.   Prove  that  there  is  no  unique  solution  of  the  system 

\     ox  +  2y  —  2z  =  0j 
[    Sx  +  ^y  -     2  =  2. 
Is  this  system  indeterminate  (cf .  Ex.  24)  ?    Explain. 

PROBLEMS 

1.  A  grain  dealer  sold  to  one  customer  5  bushels  of  wheat,  2  of  corn, 
and  3  of  rye,  for  $6.60;  to  another,  2  of  wheat,  3  of  corn,  and  5  of  rye, 
for  15.80 ;  and  to  another,  3  of  wheat,  5  of  corn,  and  2  of  rye,  for  $5.60. 
What  was  the  price  per  bushel  of  each  of  these  kinds  of  grain  ? 

2.  A  quantity  of  water,  which  is  just  sufficient  to  fill  three  jars  of  dif- 
ferent sizes,  will  fill  the  smallest  jar  exactly  4  times;  or  the  largest  jar 
twice,  with  4  gallons  to  spare ;  or  the  second  jar  3  times,  with  2  gallons 
to  spare.     What  is  the  capacity  of  each  of  these  jars? 

3.  If  A  and  B  can  do  a  certain  piece  of  work  in  10  days,  A  and  C  in 
8  days,  and  B  and  C  in  12  days,  how  long  will  it  take  each  to  do  the 
work  alone  ? 

4.  Divide  800  into  three  parts  such  that  the  first,  plus  I  of  the  second, 
plus  f  of  the  third,  shall  equal  the  second,  plus  |  of  the  first,  plus  I  of  the 
third  :  each  of  these  sums  being  400. 

5.  A  merchant  having  three  kinds  of  tea,  sold  to  one  customer  2  lb. 
of  the  first  kind,  3  of  the  second,  and  4  of  the  third,  for  $4.70;  and  to 
another  he  sold  4  lb.  of  the  first  kind,  3  of  the  second,  and  2  of  the  third, 
for  $4.-30.  If  a  pound  of  the  third  kind  is  worth  5  cents  more  than  |  lb. 
of  the  first  kind  and  I  lb.  of  the  second  kind  taken  together,  what  is  the 
price  of  each  per  pound  ? 

6.  Divide  90  into  three  parts  such  that  I  of  the  first,  plus  ^  of  the 
second,  plus  J  of  the  third,  shall  be  30 ;  and  that  the  first  part  shall  equal 
twice  the  third  part  diminished  by  twice  the  second  part. 

7.  The  sum  of  the  digits  of  a  3-digit  number  is  11 ;  the  double  of  the 
second  digit  exceeds  the  sura  of  the  first  and  third  by  1 ;  and  if  the  first 
and  second  digits  be  interchanged,  the  number  will  be  diminished  by  90. 
What  is  the  number? 

8.  The  third  digit  of  a  3-digit  number  is  as  much  larger  than  the 
second  as  the  second  is  larger  than  the  first;  if  the  number  be  divided 
by  the  sum  of  its  digits,  the  quotient  will  be  15;  and  the  number  will  be 
increased  by  396  if  the  order  of  its  digits  be  reversed.  What  is  the 
number  ? 


113-114]  SIMULTANEOUS   SIMPLE  EQUATIONS  189 

9.  The  sum  of  the  digits  of  a  4-digit  number  is  11 ;  if  the  order  of  the 
digits  be  reversed,  the  number  will  be  increased  by  819 ;  if  9  be  subtracted 
from  the  number,  the  units'  and  tens'  digits  will  be  interchanged  ;  and  the 
Slim  of  the  units'  and  tens'  digits  equals  the  hundreds'  digit.  What  is 
the  number  ? 

10.  Of  three  alloys,  the  first  contains  35  parts  of  silver,  to  5  of  copper, 
to  4  of  tin ;  the  second,  28  parts  of  silver,  to  2  of  copper,  to  3  of  tin  ;  and 
the  third,  25  parts  of  silver,  to  4  of  copper,  to  4  of  tin.  How  many  ounces 
of  each  of  these  alloys  melted  together  will  form  600  oz.  of  an  alloy  con- 
sisting of  8  parts  of  silver,  to  1  of  copper,  to  1  of  tin  ? 

11.  If  Problem  10  merely  demanded  that  the  alloy  should  contain  8 
parts  of  silver  to  1  of  copper,  how  many  ounces  of  each  of  the  given  alloys 
would  then  be  required?     Why  is  this  problem  indeterminate? 

12.  A  tank  whose  capacity  is  1600  gallons  is  supplied  by  two  pipes, 
and  has  one  outlet  pipe.  If  the  tank  is  empty,  and  all  three  pipes  are 
opened,  it  will  be  filled  in  80  hours;  if  it  is  |  full,  and  all  the  pipes  are 
opened  for  10  hours,  and  if  the  larger  supply  pipe  is  then  closed,  leaving 
the  other  two  open  10  hours  longer,  the  tank  will  then  be  |  full;  and  it 
can  be  filled  by  the  larger  pipe  alone  in  26|  hours.  Find  the  number 
of  gallons  discharged  per  hour  by  each  of  the  three  pipes,  assuming  the 
flow  to  be  uniform. 

13.  Find  an  expression  of  the  form  ax^  -\-  hx  +  c  whose  value  will  be 
6, when  x  =  2,   3  when  x  =  —  1,  and  10  when  a:  =  4. 

Suggestion.  4a  +  2^  +  cis  the  value  of  ax'^  -i-bx  +  c  when  x  =  2;  therefore. 
4a+26  +  c  =  6,  etc. 

14.  Can  such  an  expression  as  that  in  Prob..l3  be  found  which  shall 
take  four  prescribed  values  when  four  particular  values  are  assigned 
to  a:?    Why?    What  letters  represent  unknown  numbers  in  Prob.  13? 

III.    GRAPHIC  REPRESENTATION  OF  EQUATIONS* 

114.  Preliminary  remarks.  Although  an  equation  in  two  un- 
known numbers  has  an  infinitely  large  number  of  solutions, 
and  is  in  that  sense  entirely  indeterminate  (§  *101),  yet,  %  a 
beautiful  device,  due  to  a  ceiebrated  mathematician  and  philoso- 
pher named  Descartes,  a  perfectly  definite  geometric  picture  of  such 
an  equation  may  be  made.  The  "method  by  wliich  this  is  done 
will  be  explained  in  this  and  the  next  article. 

*  This  subject  is  discussed  in  detail  in  a  later  course  in  mathematics,  —  in 
Analytic  Geometry. 


190 


ELEMENTARY  ALGEBRA 


[Ch.  XI 


Y 

,Q 

P 

y' 

M 

1 

? 

y' 

Let  two  indefinite  straight  lines  X'X  and  T^T  be  drawn  at 
right  angles  to  each  other  and  intersecting  in  the  point  0 — as 

in  the  figure.  If  now  it  be  agreed  that 
distances  measured  to  the  right,  or 
upward,  be  represented  by  positive 
numbers,  Avhile  distances  to  the  left, 
or  downward,  are  represented  by  nega- 
tive numbers,  then  the  position  of  any 
point  whatever,  in  the  plane  of  this 
page,  is  completely  determined  by 
merely  giving  the  distances  of  that 
point  from  the  lines  X'X  and  Y'Y. 
It  will  be  observed  that  this  is  similar  to  locating  a  place  on  a 
map  by  means  of  its  latitude  and  longitude. 

E.g.,  to  locate  a  point  P,  whose  distances  from  I'Tand  X'X  are  respectively 
3  inches  and  2  inches,  measure  3  inches  to  the  right  from  0,  to  the  point  M  say, 
and  then  measure  2  inches  up  from  M.  This  point  is  usually  represented  by  the 
symbol  (3,  2),  i.e.,  by  P=  (3,  2) ;  the  numbers  3  and  2  are  called  the  coordinates  of 
the  point  P,  and  the  lines  X'X  and  r'Fare  called  the  axes  of  coordinates.  Simi- 
larly, the  point  Q=  (—  3,  4)  is  located  by  measuring  3  units  toward  the  left  from 
0,  and  then  4  units  upward.  The  point  R={—2,,  —  3)  is  also  represented  in  the 
figure. 

The  student  may  draw  a  figure  and  locate  accurately  the  following  points  upon 
it:*  (5,  -1),  (4,  7),  (-4,  2),  (3i,  -4),  (-2i  -5f),  and  (8,  -6|). 

115.  Geometric  picture,  or  graph,  of  an  equation.  By  the  geomet- 
ric picture  (or  map)  of  an  equation  —  usually  called  the  locus  or 
graph  of  the  equation  —  is  meant  the 
totality  of  all  those  points  whose  co- 
ordinates satisfy  that  equation.. 

E.g. ,  since  the  numbers  —  1  and  —  5,  when 
substituted  for  x  and  ?/,  respectively,  satisfy 
the  equation  2  a:  — ?/=:3,  therefore  the  point 
Pl=  (— 1,  —  5)  lies  on  the  graph  of  this  equation  ; 
so,  too,  the  points  P2=(0,  — 3),  P3  =  (l,  — 1), 
P4=(2,  1),  P5=(3,  3),  etc.,  are  on  the  graph  of 
this  equation,  because  each  of  these  pairs  of 
numbers  satisfies  the  equation. 

If  these  points  are  located,  by  the  method  of 
§  114,  it  is  found  that  they  are  not  scattered 


*  It  is  recommended  that  cross-section  paper  be  used  for  this  purpose ;  such 
paper  may  be  obtained  from  all  stationers, 


114-116]  SIMULTANEOUS   SIMPLE  EQUATIONS  191 

indiscriminately  over  the  page,  but  that  they  all  lie  upon  the  line  AB ;  this  line 
is  the  graph  of  the  given  equation.*  It  is  due  to  this  fact  that  such  equations 
are  often  called  linear  equations  (cf.  §  94). 

The  points  F^,  P3,  P4,  •••  were  found  by  assigning  the  values  0,  1,  2,  3,  -■ 
to  z,  and  then  finding  the  corresponding  values  of  y  from  the  equation;  other 
points  between  any  two  of  these  may  be  found  by  assigning  intermediate  values 
to  X. 

The  above  method  of  finding  the  graph  of  any  given  equation  in 
two  unknown  numbers  may  be  stated  thus:  by  assigning  to  ic  a 
succession  of  values,  such  as  0,  1,  2,  3,  •••,  —  1,  —2,  —3,  ••♦,  find 
the  corresponding  values  of  y,  i.e.,  find  as  many  solutions  of  the 
given  equation  as  may  be  desired ;  locate  the  points  whose  coordi- 
nates are  these  solutions,  and  draw  a  line  connecting  these  points 
in  regular  order ;  this  line  will  represent  the  required  graph. 

EXERCISES 

Draw  a  j)air  of  axes,  as  in  §§  114  and  115,  and  locate  the  following 
points : 

1.  (5,  4)  ;   (3,  7) ;  (4,  -  2) ;  (-  3,  1)  ;  and  (-  4,  -  6). 

2.  (3,  0)  ;  (-  5,  0)  ;  (0,  8)  ;  (0,  0)  ;  and  (0,  -  2). 

3.  Where  are  all  points  whose  second  number  is  0?  Where  are  those 
whose  first  number  is  0?  Where  are  all  those  whose  second  number  is 
3|?    Draw  a  fine  through  this  last  class  of  points. 

4.  Where  are  those  points  whose  second  number  is  the  same  as  its 
first  number  ?  Where  are  those  whose  second  number  is  the  opposite  of 
its  first  number?    Draw  a  line  through  each  of  these  two  classes  of  points. 

5.  What  is  meant  by  the  graph  of  an  equation  ?  Find  ten  pairs  of 
numbers,  each  of  which  satisfies  the  equation  2  x  +  y  =  12.  Carefully 
locate  the  points  determined  by  these  pairs  of  numbers. 

6.  How  many  solutions  has  such  an  equation  as  that  given  in  Ex.  5? 
Show  that  its  graph  may  be  regarded  as  a  record  of  all  of  its  solutions. 

7.  Show  that  the  equation  3  x  =  2  (i.e.,  3  x  +  0  •  2^  =  2)  is  satisfied  by 
each  of  the  following  pairs  of  numbers:  f ,  1 ;  |,  2;  |,  3;  f,  4;  etc.,  f,  0; 
I,  —  1 ;  f,  —  2 ;  etc.,  i.e.,  by  every  pair  of  numbers  of  which  the  first  is  -f . 

Where  do  all  these  points  lie  (cf.  Ex.  3)  ?  What,  then,  is  the  graph  of 
the  equation  3  a;  =  2  ?     Draw  it. 

8.  As  in  Ex.  7,  construct  the  graph  of  2  y  =  5.  Of  x-  =  -  1.  Oi  y  =  ix. 
Of  a:2  =  9. 

*  Students  who  are  acquainted  with  the  theory  of  similar  triangles  will  find  no 
dilWculty  in  proving  that  all  these  points  lie  on  the  same  straight  line  (.-l  /?),  and 
9,lso  that  the  coordinates  of  every  point  on  AB  will  satisfy  tlie  given  equation. 


192  ELEMENTARY  ALGEBRA  [Ch.  XI 

Assuming  the  graph  of  a  first  degree  equation  in  two  unknown  num- 
bers to  be  a  straight  line,  construct  the  graph  of  each  of  the  following 
equations  by  finding  two  of  its  points  and  drawing  a  straight  line  through 
them: 

9.   2x  +  y-4:  =  0.  11.   iz-y  =  d. 

10.   3y-4a:  +  2  =  0.  12.   ^  -  ^  = -?-. 

X      y      xy 

116.  Intersection  of  two  graphs.  Since  any  two  numbers  which 
satisfy  an  equation  are  the  coordinates  of  some  point  on  the  graph 
of  that  equation  (§  115),  therefore  a  pair  of  numbers  which  satis- 
fies each  of  two  given  equations  must  be  the  coordinates  of  a  point 
which  is  on  the  graph  of  each  of  these  equations,  i.e.,  these  numbers 
are  the  coordinates  of  a  point  in  which  these  graphs  intersect. 

Hence,  to  find  the  coordinates  of  the  point  in  wiiich  the  graphs 
of  two  equations  intersect  each  other,  it  is  only  necessary  to  solve 
these  equations,  regarding  them  as  simultaneous. 

On  the  other  hand,  instead  of  solving  two  simultaneous  equa- 
tions in  the  ordinary  way,  one  may  accurately  draw  the  graph  of 
each  of  these  equations,  using  the  same  axes  for  both,  and  care- 
fully measure  the  coordinates  of  their  point  of  intersection ;  these 
coordinates  will  constitute  an  approximate  solution  of  the  given 
equations. 

EXERCISES 

1.  Find  the  coordinates  of  the  point  of  intersection  of  the  graphs  of 
X  +  y  =  5  and  2  x  —  y  =  4,  both  by  solving  these  equations  and  also  by 
measurement,  and  compare  the  results. 

2.  Solve  the  system  of  equations  3  a;  +  4  ?/  =  7  and  2  x  —  S  y  =  IQ  hy 
the  graphic  method,  i.e.,  by  measuring  the  coordinates  of  the  point  in 
which  their  graphs  intersect. 

Find  the  coordinates  of  the  point  of  intersection  (as  in  Ex.  1)  of  the 
graphs  of  each  of  the  following  pairs  of  equations: 

3x-§?/  =  3,  ^      (2x-Sy=7, 

5x-7ly=n. 

6.  Show  that  the  two  equations  in  Ex.  5  are  algebraically  inconsistent. 
How  are  their  graphs  related  to  each  other  ?     Where  is  their  intersection  ? 

7.  In  how  many  points  can  two  straight  lines  intersect  each  other? 
Does  this  agree  with  §  111  ?    Explain. 


3. 


(4y-^dx=5,  ^     r3x-§?/  =  3,  ^     j 

[4x-3y  =  d.  '     [ix-2y  =  4.  '     [ 


CHAPTER   XII 
INEQUALITIES 

117.  Definitions.  Expressed  in  algebraic  language,  the  condi- 
tions of  the  problems  thus  far  met  with  have  led  to  equations; 
but  there  are  many  other  problems  whose  conditions  lead  only  to 
a  statement  that  one  of  two  expressions  is  greater  or  less  than  the 
other.  A  correct  analysis  of  such  a  statement  is  often  of  great 
importance,  and  may  afford  all  the  desired  information  concerning 
the  numbers  involved  in  the  given  problem. 

The  symbols  >  and  <  are  called  the  symbols  of  inequality,  and 
are  read  "is  greater  than,"  and  "is  less  than,"  respectively. 

Thus,  a>&  is  read  "  a  is  greater  than  6,"  and  a<  6  is  read  "a  is  less  than  b." 

One  number  is  said  to  be  greater  than  another  when  the  result 

of  subtracting  the  second  from  the  first  is  a  positive  number,  and 

one  number  is  said  to  be  less  than  another  when  the  result  of 

subtracting  the  second  from  the  first-  is  a  negative  number. 

Thus,  if  a  —  b  is  positive  then  a  >■  6,  while  if  a  —  b  is  negative,  then  a<C.b. 
Again :  since  5  —  2  =  3,  therefore  5  >  2 ;  also,  since  2  —  (—  6)  =  8,  therefore  2  >  —  6 ; 
and  since  8  —  1.5  =  —  7,  therefore  8  <<  15. 

The  statement  that  one  of  two  numbers  or  expressions  is  greater 

or  less  than  the  other  is  called  an  inequality.     The  number  or 

expression  which  stands  at  the  left  of  the  symbol  of  inequality  is 

called  the  first  member  of   the  inequality,  while  the  number  or 

expression  which  stands  at  the  right  of  this  symbol  is  called  the 

second  member,  —  the  opening  of   the  symbol  being  toward  the 

greater  number. 

Thus,  a  >  6  is  an  inequality  of  which  a  is  the  first  member  and  6  the  second ; 
it  is  read,  "  a  is  greater  than  6." 

Two  inequalities  are  said  to  be  of  the  same  species  (or  to  subsist 
in  the  same  sense)  if  the  first  member  is  the  greater  in  each,  or  if 
the  iirst  member  is  the  lesser  in  each;  otherwise  they  are  of 
opposite  species. 

Thus  the  inequalities  a  >  6  and  c  +  dy-  c  are  of  the  same  species,  while 
a-*2  +  ?/2>2:2  and  m^<in^  +  mn  are  of  opposite  species. 

193  * 


194  KLEMENTAUY  ALGEBRA  [Cii.  XII 

118.   General  principles  in  inequalities. 

(i)  //  the  same  nuinber  be  axlded  to,  or  subtracted  from, 
each  member  of  an  inequality,  the  result  will  be  an  in- 
equality of  the  same  species  as  the  given  one. 

E.g.,  10  >  8,  and  so,  also,  10  +  5  >  8  +  5,  and  10- 5  >  8  — 5. 

To  prove  this  principle  generally,  let  the  given  inequality  be  n  <  6,  and  let  c  be 
any  number  whatever ;  then  (a  +  c)  —  (6  +  c) ,  which  equals  a  —  6,  is  negative,  since 
a  <  6,  and  therefore,  by  definition, 

a  +  c  <,h -\- c. 

Similarly,  a  —  c<,h  —  c. 

Manifestly  the  proof  would  have  been  just  the  same  if  the  given  inequality  had 
been  a>6. 

From  the  principle  just  proved  it  follows  that  terms  may 
be  transposed  in  a,n  inequality,  just  as  in  an  equation,  viz., 
by  reversing  their  signs;  for  subtracting  any  given  term  from 
each  member  will  cause  that  term  to  disappear  from  one  member, 
and  to  reappear,  with  its  sign  reversed,  in  the  other. 

(ii)  //  several  inequalities  of  tJw  same  species  be  added, 
member  to  member,  the  result  will  be  an  inequality  of  the 
sajne  species. 

E.g.,  adding  the  inequalities  3  <  7,  21  <  30,  and  —  2  <  1,  member  to  member, 
we  obtain  22  <  38. 

To  prove  this  principle  generally  let  a  >  6,  c  >  d,  e  >/,  •••,  /i  >  A;  be  any  num- 
ber of  given  inequalities,  all  of  the  same  species;  then  each  of  the  differences 
a  —  6,  c  —  d,  e  — /,  •'•,h  —  k\s  positive,  hence  their  sum  is  positive, 

i.e.,  (a  —  6)  +  (c  —  cZ)  +  (e  — /)  H h  (A  —  k)  is  positive, 

hence  {a-\-c  +  e-\ (-  A)  —  (&  +  d  +/H h k)  is  positive, 

and  therefore,  a  +  c-{-e-\ \-h>h-\-d  +/H \-k;  which  was  to  be  proved. 

It  should  be  carefully  noted  that  if  two  or  more  inequalities 
which  are  not  of  the  same  species  are  added,  the  result  may  or 
may  not  be  an  inequality. 

The  student  may  illustrate  this  statement  by  means  of  some 
numerical  examples. 


118]  INEQUALITIES  195 

(iii)  //  an  inequality  he  subtracted  from  an  equation,  or 
from  an  inequality  of  opposite  species,  member  from  mem- 
her,  tim  result  will  be  an  inequality  whose  species  is  oppo- 
site to  that  of  tJie  subtrahend. 

The  proof  of  this  principle  is  similar  to  that  of  (ii)  above,  and  is  left  as  an 
exercise  for  the  student. 

The  student  may  also  illustrate,  by  appropriate  examples,  that  if  one  inequality 
be  subtracted  from  another  inequality  of  the  same  species,  the  result  may  be  an 
inequality  of  the  same  or  of  opposite  species,  or  it  may  be  an  equation. 

(iv)  If  each  mem^ber  of  an  inequality  be  multiplied  or 
divided  by  the  same  positive  number,  the  result  will  be  an 
inequality  of  the  same  species. 

E.g.,  24 >  20,  and  so,  also,  24-^  4  > 20 -^  4 ;  again,  3 <  5,  and  so  also  3  •  7  <  5  •  7. 

To  prove  this  principle,  let  a  >>  6  be  any  inequality,  and  let  c  be  any  positive 
number  whatever;  then  {a  —  b)c  is  positive,  since  each  factor  is  positive,  i.e., 
ac  —  be  is  positive,  and  hence  by  definition, 

ae  >■  6c, 
which  was  to  be  proved. 

Similarly  it  is  proved  that,  under  the  above  conditions, 

ah 
c       c 

The  principle  just  proved  enables  one  to  clear  an  inequality  of 
fractions,  and  also  to  remove  any  factors  that  are  common  to  both 
members. 

(v)  //  each  member  of  an  inequality  be  multiplied  or 
divided  by  the  same  negative  number,  the  result  will  be  an 
inequality  of  opposite  species. 

To  prove  this  principle,  let  a  >  6  be  any  inequality,  and  let  c  be  any  negative 
number  whatever;  then  {a  —  b)c  is  negative,  i.e.,  ac  —  be  is  negative,  and  hence 

ac  <  he, 

which  was  to  be  proved. 

Similarly  it  is  proved  that,  under  the  given  conditions, 

a      & 
c       c" 

(vi)  //  t?ie  signs  of  all  the  terms  of  an  inequality  be  re- 
versed, then  the  symbol  of  inequality  must  also  be  reversed. 

E.g.,  if2a  —  4c'  +  3a;>2d  +  5y  —  7  6,  then  4c  —  2a-3a;<76-2d  —  5y. 
The  proof  of  this  principle  follows  directly  from  (v)  by  putting  —  1  for  the 
multiplier  c. 


196  ^  ELEMENTARY  ALGEBRA  [Ch.  XII 

(vii)  If  the  first  of  three  numhers  is  greater  than  the 
second,  and  the  second  is  greater  than  the  third,  then  the 
first  is  greater  than  the  third;  and  conversely. 

E.g.,  10  >  7  and  7  >  3,  and  10  >  3  also. 

To  prove  this  principle,  let  a  >  &  and  6  >  c  be  the  given  inequalities ;  then 
a  —  6  is  positive,  as  is  also  6  —  c,  and  hence  their  sum  (a  —  h)  +  (6  —  c),  i.e.,  a  —  c, 
is  positive,  and  therefore  a  >  c,  which  was  to  be  proved. 

Similarly  it  is  proved  that  if  a  <  &  and  6  <  c,  then  a  <  c. 

(viii)  If  two  inequalities  which  are  of  the  same  species, 
and  whose  members  are  all  positive,  he  multiplied  togetJier, 
meinher  by  member,  the  result  will  be  an  inequality  of  the 
same  species. 

E.g.,  5  >  3  and  4  >  2,  and  5  . 4  >  3  •  2  also. 

To  prove  this  principle,  let  a  >  6  and  c  >  (^  be  two  such  inequalities ;  then  by 
(iv)  ac  >  be,  but  by  (iv)  he  >  hd,  whence  by  (vii)  ac  >  hd,  which  was  to  be  proved. 

By  proceeding  step  by  step,  it  is  clear  that  principle  (viii)  holds  for  any  num- 
ber of  (and  not  merely  for  two)  such  inequalities. 

The  student  may  modify  the  above  statement  and  proof  so  as 
to  apply  to  the  case  in  which  some  of  the  members  are  negative. 


EXERCISES 

1.  When  is  the  first  of  two  numbers  said  to  be  greater  than  the 
second  ?     When  is  it  said  to  be  less  ? 

2.  By  the  definitions  of  "greater"  and  "less  "  given  in  §  117,  show 
that  5  >  2 ;  that  -  23  <-  12 ;  and  that  2  >  -  9. 

3.  li  a=^  b,  show  that  a"^  +  b^>2  ah.  This  is  a  very  important  rela- 
tion, and  well  worth  remembering. 

Suggestion,    (a  —  6)2  is  positive  whether  a  >  &  or  a  <  6. 

4.  If  two  or  more  inequalities  of  the  same  species  are  added,  what  is 
the  species  of  the  resnlting  inequality?  Prove  your  answer.  Is  it  neces- 
sary that  the  members  of  these  inequalities  hQ positive  numbers? 

5.  If  an  inequality  is  subtracted  from  another  inequality  of  the  same 
species,  member  from  member,  what  is  the  result?    Prove  your  answer. 

6.  If  two  inequalities  of  the  same  species  are  multiplied  together, 
member  by  member,  what  is  the  result?  Prove  your  answer.  Is  it 
necessary  in  this  case  that  the  members  of  these  inequalities  be  positive 
numbers? 


118-119]  INEQUALITIES  197 

7.  What  happens  if  the  signs  of  the  terms  of  each  member  of  an 
inequality  are  reversed?     Why? 

8.  May  terms  be  transposed  from  one  member  of  an  inequality  to 
the  other  ?    If  so,  how  and  why  ? 

9.  What  other  operations  may  be  performed  with  or  upon  inequali- 
ties, producing  results  whose  relations  are  known  ? 

10.  Name  and  illustrate  some  operations  with  inequalities  that  give 
results  about  whose  relations  there  is  doubt.  E.g.,  the  quotient  of  two 
inequalities  of  the  same  species,  divided  member  by  member,  may  be  an 
equality  or  an  inequality  of  the  same  or  of  opposite  species. 

119.  Unconditional  and  conditional  inequalities.  An  unconditional 
inequality  is  one  which  is  true  for  all  values  of  the  letters  in- 
volved—  e.g.,  a  +  4>a;  while  a  conditional  inequality  is  one 
which  is  true  only  on  condition  that  the  values  to  be  assigned 
to  the  letters  involved  shall  be  somewhat  restricted  —  e.g., 
a;  +  4<3a;  —  2  only  on  condition  that  the  values  assigned  to  x 
shall  be  greater  than  3.* 

To  solve  a  conditional  inequality  means  to  find  those  values  of 
its  letters  for  which  the  inequality  is  true ;  this  may  be  done  by 
means  of  the  principles  which  were  proved  in  the  preceding 
article  = —  for  illustrations  see  Exs.  1  and  2  which  follow. 

Ex.  1.   Given  3  x  —  ^^  >  ^^  —  ar,  to  find  the  possible  values  of  x. 

Solution.     On   multiplying  each  member  of  the  given  inequality 

by  3,  it  becomes 
^  9  a:  -  25  >  11  -  3  a:,  [§118  (iv) 

whence  9  a:  +  3  a;  >  11  +  25,  [§  118    (i) 

i.e.,  V2x>  36, 

whence  a:>3;.  [§  118  (iv) 

therefore,  if  the  given  inequality  is  true,  x  must  be  greater  than  3. 

By  means  of  the  principles  established  in  §  118  the  student  may  show  that  each 
step  in  the  reasoning  of  Ex.  1  is  reversible,  and  hence  that  the.  converse  of  that 
example  is  also  true ;  viz.,  that  if  x  >■  3,  then  3  a;  —  -^/-  >►  ^-^-  —  x. 

*  Let  it  be  observed  that  conditional  and  unconditional  inequalities  are  respec- 
tively analogous  to  conditional  and  identical  equations;  the  student  may  also 
note  the  analogy  between  solving  an  inequality  and  solving  an  equation. 


198  ELEMENTARY  ALGEBRA  [Cri.  XII 

"ig  I'  ^°  ^°^  those  values  of 

x  and  y  that  will  satisfy  them  both. 

Solution.  On  multiplying  each  member  of  the  inequality  by  4,  and 
each  member  of  the  equation  by  3,  they  become,  respectively, 

8x+12?/>20, 
and  3a:+123/  =  18; 

whence,  subtracting,  5a:>2,  [§118    (i) 

and  therefore  a:>|.  [§  118  (iv) 

Now  substitute  for^a:  any  number  greater  than  g,  in  the  above  equation, 
and  find  the  corresponding  value  of  y ;  these  values  of  x  and  ?/,  taken 
together,  will  satisfy  both  the  equation  and  the  inequality. 

EXERCISES 

3.  Distinguish  between  a  conditional,  and  an  unconditional  inequality. 
To  which  of  these  classes  does  aP-  +  h'^  +  \>2  ah  belong?     Why  ? 

4.  Is  the  expression  6x  —  5>3x  +  10  true  for  all  values  of  x ?  If  not, 
what  is  the  least  value  that  x  may  have  in  this  inequality  ?  To  which 
class  does  this  inequality  belong? 

5.  What  is  meant  by  "  solving  "  a  conditional  inequality  ?  Describe 
the  procedure.  Illustrate  what  you  have  said  by  solving  the  inequality 
in  Ex.  4. 

6.  From  the  inequality  in  Ex.  4  above  it  is  found  that  a:  >  5,  i.e.,  the 
range  of  values  that  x  may  have  in  this  inequality  is  from  just  above  5 
upward ;  5  may  here  be  called  the  lower  limits  or  minimum,  of  the  possible 
values  of  x.     Find  the  minimum  value  of  a:  in  3  a;  <  5  a;  —  9. 

7.  Show  that  the  range  of  values  of  a:  in  a;^  +  24  <  11  a;  is  between  3 
and  8,  i.e.,  that  3  is  the  lower  limit,  or  minimum,  and  that  8  is  the  upper 
limit  or  maximum. 

Suggestion.  In  order  that  {z  —  3)  (8  —  a;) ,  i.e.,  Wx  —  x"^  —  24,  may  be  positive, 
both  factors  must  be  positive  or  both  negative. 

Find  the  range  of  values  of  x  in  each  of  the  following  inequalities : 

8.  a:2  >  9.  13.   x'^  +  o  a:  >  24. 

9.  .2  +  24>lla:.  ,  J4.-ll>-, 

10.  30>a:+-^>25.  ^*-    |io_x>5.^ 

11.  28>3a:  +  a;2.  j3-4r<7, 

12.  a:2>9a:-18.  '     la:+2<4. 


119]  INEQUALITIES  199 

16.  By  the  definitions  of  "greater"  and  "less"  given  in  §  117,  show 

that  n  +  -<2,  when  n  is  any  positive  number,*  i.e.,  show  that  the  sum  of 
n 

any  positive  number  and  its  reciprocal  is  not  less  than  2. 

17.  Show  that  4  x^  +  9  <  12  x.* 

18.  Show  that26(6a-5  6)X2a  +  &)(2a-6). 

If  a,  h,  and  c  are  positive  and  unequal,  prove  the  correctness  of  the  fol- 
lowing statements: 

19.  a2  +  62  ^  c2>>  ab  +  hc  ■\-  ac. 

20.  a^-\-b^>  a%  +  abK  21.   a^  +  68  +  cS  >  3  abc, 

22.  If  a2  +  62  =  1,  and  c"^  +  cP  =  1,  prove  that  a6  +  erf  >  1.* 

23.  If  m  and  n  are  both  positive,  which  of  the  expressions  ^  ^^  or 
^'"'^    is  the  greater? 


m  '\-n 

Solve  the  following  systems : 

r2x-3j/<2,  f3a:+2y  =  42,  fx  +  y=10, 


24. 


3  3^. 


l2x  +  52/=:6.  |3a:-|>16.  l4:c< 

I  20      15  ^ 

Find  the  integral  values  of  x  and  y  in  the  following  systems : 

l3x-2/<21.  |l3a:-^<33. 

31.  If  16  more  than  3  times  the  number  of  sheep  in  a  certain  flock 
exceeds  27  plus  twice  their  number,  and  if  45  less  than  4  times  their 
number  is  less  than  their  number  diminished  by  6,  how  many  sheep  are 
there  in  the  flock  ? 

32.  Find  the  smallest  integer  fulfilling  the  condition  that  \  of  it 
decreased  by  7  is  greater  than  \  of  it  increased  by  6. 

33.  Find  a  simple  fraction  (in  its  lowest  terms)  which,  when  2  is  added 
to  its  numerator  and  subtracted  from  its  denominator,  shall  be  greater 
than  f,  while  if  2  is  subtracted  from  its  numerator  and  added  to  its 
denominator,  it  shall  be  less  than  \. 

34.  Three  times  A*s  money  and  4  times  B's  is  $  1  more  than  6  times 
A's ;  and  if  A  gives  $  5  to  B,  then  B  will  have  more  than  6  times  as  much 
as  A  will  have  left.     Find  the  range  of  values  of  A's  money  and  B's. 

♦Compare  also  Ex.  3,  p.  196.    The  symbol  <[  stands  for  "is  not  less  than." 


200  ELEMENTARY  ALGEBRA  [Cii.  XII 


REVIEW  QUESTIONS- CHAPTERS  XXII 

1.  Define  and  illustrate:  conditional  equations;  equivalent  equations; 
integral  equations ;  the  degree  of  an  equation ;  literal  equations. 

2.  Outline  the  plan  for  solving  a  conditional  equation  in  one  unknown 
number,  and  state  the  principles  upon  which  this  plan  rests. 

3.  How  may  a  fractional  equation  in  one  unknown  number  be  solved? 

4.  Under  what  circumstances  are  extraneous  roots  introduced  by- 
clearing  an  equation  of  fractions?      How  may  such  roots  be  detected? 

5.  By  means  of  the  equation \-  — — — -  =  -^^ — -^,  illus- 

trate  your  answer  to  Lk.  4. 

6.  Define  and  illustrate  what  is  meant  by  :  an  indeterminate  equa- 
tion; an  indeterminate  system  of  equations;  consistent  equations;  inde- 
pendent equations ;  simultaneous  equations. 

7.  Outline  three  methods  of  elimination. 

8.  Prove  that  the  syetem  of  equations  a^x  +  b^y  =  Cj  and  Og^;  +  &22/  =  Cg 
has  one  solution,  and  only  one,  if  a^b^  ^  «2*i- 

9.  Outline  the  procedure  for  solving  a  system  consisting  of  n  inde- 
pendent simple  equations  in  n  unknown  numbers. 

10.  Find  an  expression  of  the  form  ax^  +  bx  +  -   whose  value  is  16 
when  X  =  —  1,  2  when  x  =  1,  and  40  when  x  =  2.      ^ 

11.  What  is  meant  by  the  graph  of  an  equation?    Illustrate  your 
answer. 

12.  How  may  the  graph  of  an  equation  be  constructed  ?    Construct  the 
graph  oi  6y  =  Sx  -\-  lO;  also  of  2 y^  =  S x  -\-  1. 

13.  How  may  a  pair  of  equations,  such  as  that  given  in  Ex.  8,  be  solved 
graphically  ?    Illustrate  your  answer. 

14.  Define  a  conditional  inequality,  also  an  unconditional  inequality. 
Illustrate  each. 

15.  How  may  a  conditional  inequality  be  solved?     Illustrate  your 

answer  by  finding  the  range  of  values  of  x  in  the  inequality  x  —  3  <  — . 

10  '"' 

16.  If  X  -  3  <  — ,  does  it  follow  that  a:^  -  3  x  <  10  ? 

X 

17.  Prove  that  a  positive  proper  fraction  is  increased  by  adding  the 
same  positive  number  to  both  its  numerator  and  its  denominator. 


CHAPTER   XIII 

INVOLUTION  AND  EVOLUTION 

I.    INVOLUTION 

120.  Definitions.  If  a  represents  any  number  *  whatever,  then 
it  has  been  agreed  that  the  product  «•«•«•••  (to  n  factors), 
which  is  called  the  /ith  power  of  a,  shall,  for  brevity,  be  represented 
by  the  symbol  a",  which  is  usually  read  "a  nth."  The  number 
a  is  called  the  base,  and  n  the  exponent,  of  the  power  [cf.  §  7  (iv)]. 

The  operation  of  raising  a  number  to  any  given  power  is  called 
involution.  It  consists  merely  in  a  succession  of  multiplications; 
thus,  43  =  4.4.4  =  64,  (-2/  =  -32,  (a+&)2  =  a2  +  2a6  +  6^  etc. 

Under  the  above  definition  the  symbol  a"  has  been  appropriated 
only  when  71  is  a  positive  integer ;  that  definition  assigns  no  mean- 
ing whatever  to  such  expressions  as  a~^,  a°,  and  a^.  In  §  44 1  it 
was  shown,  however,  that  in  operating  with  such  symbols  as  a"  it 
is  often  advantageous  to  make  the  further  agreement  that  a"*, 

where  k  is  any  positive  integer,  shall  mean  — ,  and  that  a"  shall 

mean  1.     In  Chap.  XIV  such  symbols  as  a^  will  have  a  meaning 
assigned  to  them,  and  will  receive  detailed  consideration. 

121.  The  exponent  laws.  Under  the  above  agreements  as  to  the 
meaning  of  a",  the  following  laws  for  exponents  are  easily  estab- 
lished. 

(i)  First  exponent  law.  If  a  is  any  base,  and  m  and  n  are 
integers  (positive  or  negative),  or  zero,  then 


*  The  word  number  is  here  used  to  include  algebraic  expression  also, 
t  This  article  should  now  be  reread.  J  Compare  also  §  37. 

201 


202  ELEMENTARY  ALGEBRA  [Ch.  XIII 

For,  if  m  and  n  are  positive  integers,  then 
a"*  •  a"  =  (a  •  a  •  a  •••  to  m  factors)  •  (a  •  a  •  a  •••  to  n  factors) 
=z  a  '  a  '  a  '•  •  to  (m  -\-  n)  factors  [Associative  law 

=  a'"+". 

If  either  m  or  n  is  a  negative  integer,  say  n  =  —  Jc,  where  A;  is  a 
positive  integer,  then 

a*  a* 

_  g  •  g  '  g  • '  •  to  m  factors 
g  .  g  .  g  •••  to  k  factors 

g*  "• 
according  as  m  >  A;,  or  m  <  A; ; 

but  (since  ri  =  —  fc)      g*""*  =  «"*+'*, 

and  —  =  g-^*-"*)  =  g*""*  =  g^^- : 

therefore  g"* .  g"  =  g'"+", 

even  if  one  of  the  exponents  is  a  negative  integer. 

Similarly  the  student  may  prove  the  correctness  of  this  law  if 
both  m  and  n  are  negative,  or  if  either  or  both  of  them  are  0. 

By  successive  applications  of  the  foregoing  law,  and  with  the 
same  limitations  upon  the  exponents,  it  follows  that 

a"'  •  a"  •  a^  •  a''  -  =  fl^«+n+p+'-+-. 

(ii)  Second  exponent  law.     If  a  is  any  base,  and  m  and  n  are 
integers  (positive  or  negative),  or  zero,  then 

For,  if  m  and  n  are  positive  integers,  then 
(g*")"  =  (g  •  g  .  g  ...  to  m  factors)" 

=  g  •  g  •  g  •  •  •  to  mn  factors  [Associative  law 


121]  INVOLUTION  AND  EVOLUTION  203 

and  if  either  m  or  n  is  a  negative  integer,  say  m  =  —  k^  where  k 
is  a  positive  integer,  then 

(«"•)«  =  (a-*^)"  =  /^i-Y=—  '  —  '  —  '"  ton  factors 

= -J— =  —  =  a-*^"  =  a"*".        r—k==m 

If  both  m  and  n  are  negative,  or  if  either  or  both  of  them  are 
zero,  the  proof  is  similar  to  that  just  given;  hence,  for  all  these 
cases,  ^^r.y  ^  ^«„ 


(iii)  Third  exponent  law.     If  a  and  h  are  any  two  bases,  and 

n  is  a  positive  or  negative  integer,  or  zero,  then 

al"  •})''  =  {obY. 
For,  if  n  is  a  positive  integer,  then 

a**  •  6"  =  (a  •  a  •  a  •••  to  n  factors)  .(p'b-b-"  to  n  factors) 

=  a6  .  a5  .  a5  . . .  to  n  factors  [Commutative  and 

[_  associative  laws 
=  (aby; 

if  n  is  a  negative  integer,  say  n  =  —  k,  where  A;  is  a  positive  inte- 
ger, then 

a"  .  6"  =  a-* .  6-*  =  i  ^^  =  "Fir.  =  7^7*  =  (^^)"  =  («^)^*' 
a*     b''     a*  •  &*     {aby 

as  before  ; 

and  if  n  =  0,  then  a"  •  5"  =  1  =  (aby ;  [Since  a;^'  =  1 

hence,  for  all  these  cases,  a"  •  5**  =  (aby. 

By  successive  applications  of  the  above  law  it  follows  that 

aj'lf  c"^'  d^  •••  =  {ohcd  — )"• 

(iv)  Fourth  exponent  law.     If  a  is  any  base  and  m  and  n  are 
any  integers,  or  zero,  then 

a""  ^  a""-  a^'*. 


204  ELEMENTARY  ALGEBRA  [Ch.  XIII 

The  proof  of  the  correctness  of  this  law  rests  directly  upon  the 
first  exponent  law  [(i)  above],  and  the  definition  of  a  quotient 
[§  3  (iv)],  for,  since       ^m-n  ,  ^^n  ^  ^m-n+n  ^  ^r.^  [-0  above 

therefore  .  a"'  ^  a"  =  a"*-".  [§  3  (iv) 

EXERCISES 

1.  Write  a  carefully  worded  statement  of  each  of  the  four  exponent 
laws  above,  —  e.g.,  the  third  law  may  be  stated  thus :  "  The  product  of 
like  powers  of  any  two  or  more  numbers  is  the  like  power  of  the  product 
of  those  numbers." 

2.  How  is  the  sign  of  the  power  in  such  a  case  as  (—  6^)5  determined? 
State,  illustrate,  and  prove  a  law  which  shall  cover  all  such  cases,  bearing 
in  mind  that  the  exponents  may  be  positive  or  negative  integers,  or  zero 
(cf.  §  18,  especially  note  2). 

3.  Tell  what  the  ngn  of  the  result  in  each  of  the  following  expres- 
sions is,  and  explain  your  answer : 

(-a)3;  (-a)^  (a)-^;  (- a)-3;  (-4)0;  (-f^'V';  (-6)^80; 
(-a:)2";    and    (-2)2«-i.  ^     ' 

What  is  the  value  of  (-  2)3  .  2-2?    Of  3-2  •  (-  2)3?    Of  32  .  2-2? 

4.  How  is  a  fraction  raised  to  a  power  ?  Why  ?  Give  four  illustra- 
tions of  your  answer.     Read  again  the  second  paragraph  of  §  120. 

5.  What  does  a  represent  in  the  proofs  of  §  121?  May  it  represent 
any  polynomial  whatever,  as  well  as  any  number  ?  What  does  it  repre- 
sent in  Ex.  3  ? 

6.  By  §  62  expand  the  following  expressions :  {x  +  y)^  {x  +  y)^, 
and  {x  +  z/)^;  then  multiply  the  first  two  expanded  forms  together,  and 
thus  verify  that  {x  +  yY  •  {x  +  yY  =  (x  +  y^- 

7.  To  what  kind  of  numbers  were  exponents  originally  limited  ?  To 
what  extent  has  this  limitation  now  been  removed  ?  What  is  the  meaning 
of  such  an  expression  as  x~^  ?  Of  x^  ?  Read  again  §  44,  and  the  third 
paragraph  of  §  120. 

Simplify  the  following  expressions  (free  them  from  negative  and  zero 
exponents  where  such  occur,  etc.)  and  explain  each  step  of  your  work 
fully,  always  referring  to  the  appropriate  exponent  laws : 

8.  a^Wc^\  a-3i-%-3;  and  (a-2)8. 

9.  (a2x8^-2)4;     (m3^?/-4)2.       ^nd    (rt2y-3)-2. 

10.  (rt2:^2)3_^  (_a^2)2.  (6a0)2  -  (2x0)2;  and  (- 122a-8j:V)2 -^ 
(-32a-5x2)3. 


121-122]  INVOLUTION  AND   EVOLUTION  205 

12.    f--^y.  14.    l-^^J^!n\\  16     /_zr!£!^V". 

17.  State  the  binomial  theorem  (§62). 

18.  How  many  terms  are  there  in  the  expansion  of  (m  +  n)^?  How- 
many  in  (a  -  6)8?     How  many  in  (3  s  -  2  0"? 

19.  What  are  the  signs  of  the  terms  in  (a  -  &)8?  Compare  (a  -  by 
with  [a  +  (-  b)y,  and  explain  why  the  alternate  terms  of  the  expansion 
are  negative. 

Write  down  the  expansions  of  the  following  expressions,  and  remove 
negative  exponents  where  they  present  themselves  : 

20.  (2a-36)4.t  25.  (a-hb-hcy,i.e.,[a+(b  +  c)f. 

21.  (x2-i/t)8.  26.  (3xY-^fz^y- 

22.  (x-2  +  3  z/-i)4.  27.  (a;3  -  2  ?/-3)5. 

23.  (3  a  +  2  &2)5.  28.  (a-2  -  ic-i)^. 

24.  (2  m +3  xy.  29.  (2  x8  +  3  a;2  -  5)*. 

30.  Is  (a- b'C-  <iy  equal  to  a^  •  b"^  •  c^  ■  d^'i  is  (a  +  b  +  c  +  dy  equal 
to  a^  +  62  _^  ^2  _|_  ^2  9     Explain  your  answer. 

Is  involution  distributive  over  a  product  (cf .  §  39)  ?  over  a  sum  ? 

31.  Translate  the  following  symbolic  statement  into  a  verbal  one : 

(a  +  a;)«  :^  a«  +  x". 

32.  Ts  [(-2)3]3  equal  to  [(-2)2]3?  What  is  the  sign  of  each 
result?    Why? 

33.  Prove  that  (a"*)"  =  (a")'",  wherein  a  is  any  number  or  algebraic 
expression,  and  m  and  n  are  integers  (positive  or  negative)  or  zero  [cf. 
law^  (ii)  above].     Also  state  this  principle  in  words. 

II.   EVOLUTION 

122.  Definitions.  A  number  whose  nth  power  is  a  given  num- 
ber {n  being  any  positive  integer)  is  called  an  nth  root  of  the 
given  number ;  thus,  if  a"  =  h,  then  a  is  an  ?ith  root  of  h.  % 

*  Compare  Exs.  20-26,  §  93.  t  Compare  note,  §  57. 

J  As  here  used  the  word  number  mcludes  algebraic  expression  also. 


206  ELEMENTARY  ALGEBRA  [Ch.  XIII 

E.g.,  2  is  a  3d  root  of  8  because  2^  =  8 ;  so  also  2  ab^  is  a  5th  root  of  32  a^ftW ; 
either  +3  or  —3  {i.e.,  ±3)  is  a  2d  root  of  9;  db|  is  a  4th  root  of  if ;  a  3d  root  of 
x8  —  3  a;2y  +  3  xy^  —  y^  is  x  —  y;  etc. 

The  special  names  square  root  and  cube  root  are  usually  employed  instead  of 
2d  root  and  3d  root,  respectively  [cf.  §  7  (iv),  note]. 

The  operation  of  finding  any  root  of  a  given  number  is  called 
evolution,  or  extraction  of  roots.  Evolution  is  then  the  inverse  of 
involution,*  just  as  subtraction  is  the  inverse  of  addition,  and 
division  the  inverse  of  multiplication. 

The  radical  sign,  ^,  is  placed  before  a  number  to  indicate  that 
a  root  of  the  given  number  is  required,  and  a  small  figure,  called 
the  index  of  the  root,  is  placed  in  the  opening  of  the  radical  sign 
to  indicate  what  particular  root  is  to  be  extracted. 

The  number  whose  root  is  required  is  called  the  radicand ;  and 
an  indicated  root  is  said  to  be  an  even  root  or  an  odd  root  accord- 
ing as  its  index  is  an  even  or  an  odd  number. 

Thus,  y/21  stands  for  the  cube  root  of  27 ;  this  is  an  odd  root  since  its  index,  3, 
is  an  odd  number,  and  27  is  the  radicand.  \/(32  a^&io)  jg  the  5th  root  of  32  a^ftio, 
and  y/a  is  the  nth  root  of  a.  If  no  index  is  written,  the  index  is  understood 
to  be  2,  i.e.,  \/4  stands  for  the  square  root  of  4. 

The  radical  sign  is  a  modification  of  the  letter  r — the  initial  letter  of  the  Latin 
word  radix,  meaning  root. 

In  practice  the  radical  sign  is  usually  combined  with  a  vinculum  (§  8)  to  indi- 
cate clearly  just  how  much  of  the  expression  following  the  radical  sign  is 
to  be  affected  by  that  sign  ;  thus  V9  +  16  means  the  square  root  of  the  sum  of  9 
and  16,  while  y/^-\- 16  indicates  that  16  is  to  be  added  to  the  square  root  of  9. 

Instead  of  the  vinculum  a  parenthesis  may  be  used  for  the  same  purpose,  in 
connection  with  a  radical  sign,  thus :  \/(9+16)  =  \/9+16,  VaSfts .  c=  y/{<ofib^)  •  c,  etc. 

123.  Roots  of  monomials.  If  a  monomial  is  an  exact  power,  the 
corresponding  root  can  usually  be  written  down  by  inspection. 

E.g.,  v'8 a6a;3  =  2 a'^x,  because  (2 a2cc)8=  8 a^xS  (§  121);  V9xV=  +  3a;V  or 
—  3a;V,  because  (+  3  xV)^  =  (—  3  a;%8)2  =  9  a;4^y6;  ^—32x10  =  —  2x2,  because 
(_ 2x2)6  =-32x10;  \l^  =  '^Jnk  because  f2m\3^8mf  etc  See  also  Exs.  5 
and  21  below. 


*  It  is  to  be  remarked,  however,  that  while  raising  a  number  to  a  power  always 
produces  a  single  result,  extracting  a  root  may  lead  to  more  than  one  result; 
e.g.,  32  =  9,  but  the  square  root  of  9  =+  3  or  —  3. 

This  is  often  expressed  by  saying  that  involution  is  a  unique  operation,  while 
evolution  is  non-unique. 


122-123]  INVOLUTION  AND  EVOLUTION  207 

EXERCISES 

1.  What  is  meant  by  the  square  root  of  a  number?  Of  what  two 
equal  positive  factors  is  25  the  product?  What,  then,  is  a  square  root  of 
25  ?     Has  25  another  square  root  ?     Why  ? 

2.  What  are  the  square  roots  of  49  ?  Why  ?  The  fourth  roots  of  81  ? 
Why?  Prove  that  if  a  is  any  even  root  of  a  number,  then  —  a  is  also  a 
root  (with  the  same  index)  of  that  number. 

3.  What  is  the  cube  root  of  27?  W^hy?  Of  -  27  ?  Why?  Of  64 
and  of  -  64?     Why?    How  does  ^32  compare  with  \/^r82?     Why? 

Compare  the  signs  of  odd  roots  of  numbers  with  the  signs  of  the 
numbers  themselves,  and  give  your  reasons  in  full.  Is  this  also  true 
for  even  roots? 

4.  What  is  the  sign  of  any  even  power  of  any  positive  or  negative 
number?  Why?  Can,  then,  an  even  root  of  a  negative  number  be  an 
integer  or  a  fraction,  positive  or  negative  ?     Why  ? 

5.  What  is  the  nth  power  of  a^S^j-ft^-s?  What,  then,  is  \^3nj2n-cftny"-5n  ? 
Why  ?  What  is  the  sign  of  this  root  ?  Why  ?  How  do  the  exponents  of 
the  root  compare  with  those  of  the  number  itself?     Why? 


6.  Is  V9.16equalto\/9.\/l6?  Why?  Is  V9+16equalto  \/9+ Vl6? 
Compare  Ex.  30,  §  121,  and  give  a  verbal  statement  of  your  general  con- 
clusion. 

Find  the  following  indicated  roots,  and  verify  your  answers.  Also  tell 
which  are  even  and  which  are  odd  roots,  and  name  the  radicand  and  the 
index  in  each  case  : 


7. 

^a366ci6. 

8. 

VI 6  a^x^y-'^. 

9. 

>/32  x6?/io. 

10. 

Va^"x-^y^\ 

11. 

f-? 

13.    v/128  a^^ft-i^y. 


8  /  256  yn  V« 
■    \6561^3%-8 


j_^     3  /     125  a;i2yg 

'    V      1728  a^z^' 


18. 


.027  a'\r^ 


'    \      128a:i4 


15.     I     Hi'^-y)      .  „        2na2n2-2^ 


19. 


\ 


_     5  I-  32  «5:,40  I, 


a5xjjx2^dx 


12.    V-  243  a^^x-^  \    243  y^s  \  2-^^y^'z' 

21.  Write  a  rule  for  the  extraction  of  such  roots  as  the  above,  and 
emphasize  particularly  the  matter  of  exponents  and  signs.  Does  your 
rule  apply  to  roots  of  polynomials  also  ? 


208  ELEMENTARY  ALGEBRA  [Ch.  XIII 

124.  Roots  of  polynomials  extracted  by  inspection.  If  a  poly- 
nomial is  an  exact  power  of  a  binomial,  a  little  study  will  usually 
reveal  the  corresponding  root ;  this  is  illustrated  by  the  following 
examples. 

Ex.  1.     Find  the  square  root  of  m^  +  4  m%  +  4  n^. 

Solution.  This  expression  is  easily  seen  to  be  (m^  +  2n)2;  there- 
fore Vm*^  +  4  m^n  +  4  n^  =  i  (m^  +  2  n). 

Ex.  2.  Find  the  cube  root  of  8  a^  _  36  a%  -  27  b^ -\-  54  ab^ 
Solution.  Since  the  given  polynomial  has  four  terms,  two  of  which, 
viz.,  8  a^  and  —  27  h^,  are  exact  cubes,  therefore  it  may  be  the  cube  of  a 
binomial  (§  62)  ;  if  it  is  the  cube  of  a  binomial,  that  binomial  must  be 
2  a  —  3  6  (why  ?),  which,  on  further  examination,  proves  to  be  the  required 
cube  root. 

Hence  v'S  a^  -  36  a%  -  27  6^  +  54  ab^  =  2  a -8b. 

A  polynomial  which  is  the  square  of  another  polynomial  may 
also  sometimes  be  recognized  as  such  (cf.  §  61),  and  its  square 
root  may  then  be  written  down  by  inspection. 

Ex.  3.     Find  the  square  root  of  a^  +  62  _  2  aJ  -  4  &c  +  4  c^  +  4  ac. 

Solution.  Since  the  given  polynomial  consists  of  six  terms,  three  of 
which  are  exact  squares,  and  three  of  which  are  double  products,  there- 
fore (§  61)  it  mmj  be  the  square  of  a  trinomial  whose  terms  are  the  square 
roots  of  the  square  terms ;  by  a  little  further  examination  it  is  seen  that 

Va2  +  62  _  2  a6  -  4  &c  +  4  c2  +  4  ac  =±(a-b  +  2c). 

EXERCISES 

Extract  the  following  indicated  roots  by  inspection,  and  verify : 
4.    ^4  x2  +  12  a:  +  9.  6.    V(m  +  n)2  -4:(7n  +  n)+  4. 


5.    V25  2/2  _  40  y  +  16.  7.    Vx^  +  2xy  +  y^ -2xz -2yz  +  z^. 

8.  \/8  h^  -  84  h%  +  294  hk^  -  343  k\ 

9.  \/x^  -^xhj  +  y^-^xy^  +  Q  x'^y\ 


10.  a/8  w8  _  12  u^v  -  v3  +  6  uv\ 

11.  v^flS  _  65  _  5  ^4^,  +  5  aj4  +  10  rtS^'i  _  10  an\ 

12.  Va2  +  9  62  _  6  a6  +  6  (x  -  2  2/)  (a  -  3  6)  +  9  (a:2  -  4  a:?/  +  4  y'^). 

13.  Vx^  -  6  abx^  +  15  a%^x^  -  20  a%^x^  +  15  a'^b^x'^  -  6  a%^x  -f-  a%^. 


124-125]  INVOLUTION  AND  EVOLUTION  209 

125.  Square  roots  of  polynomials.  Since  it  is  not  always  easy 
to  lind  the  square  root  of  a  polynomial  by  the  method  illustrated 
in  §  124,  another  method,  which  is  always  applicable,  will  now  be 
given.  This  method  will  be  better  understood  by  first  squaring 
a  polynomial  and  carefully  observing  its  formation,  and  then 
reversing  that  process. 

(i)  Consider  first  the  binomial  A-\-B;  its  square  is,  A^  +  2AB-\-  B^,  therefore 
the  square  root  of  A^  +  2  AB  +  B'^  is  J.  +  jB  ;  and  the  question  now  to  be  investi- 
gated is:  given  the  power  A^  +  2  AB-\- B^,  how  may  the  root  A-\-B  he  found 
from  it? 

Since  the  first  term  of  the  power  is  the  square  of  the  first  term  of  the  root, 
therefore  the  first  term  of  the  root  is  the_square  root  of  the  first  term  of  the 
power ;  i.e.,  the  first  term  of  the  root  is  \A^,  viz.,  A.* 

If  the  square  of  the  root  term  just  found  be  subtracted  from  the  given  power, 
then  the  first  term  of  the  remainder,  viz.,  2AB,  will  be  the  double  product  of  the 
first  and  second  terms  of  the  root,  therefore  the  second  term  of  the  root  is  found 
by  dividing  the  first  term  of  the  remainder  by  twice  the  root  already  found. 

Twice  the  root  already  found  at  any  stage  of  the  work  is  usually  called  the 
trial  divisor,  and  the  trial  divisor  plus  the  next  root  term  is  called  the  complete 
divisor. 

The  work  of  finding  the  square  root  just  considered  may  be  put  into  the  fol- 
lowing form : 

^2  +  2^5  +  ^2     1^  +  ^ 

^2 


Trial  divisor,  2  A 

Complete  divisor,  2  A 


2AB-{-B^ 

2AB  +  B^         =i2A  +  B)'B 


Observe  that  the  first  and  second  subtractions  are  together  equivalent  to  the 
subtraction  of  (A  +  B)^  from  the  given  power. 

Similarly,  to  find  the  square  root  of  9  m2  —  42  mx^  +  49  x^,  the  work  may  be 
arranged  thus  (the  student  should  fully  explain  each  step  of  the  process) : 


9  m2  —  42  mx^  +  49  x^     \Sm  —  7x^ 
9m2 


Trial  divisor,  6  m 

Complete  divisor,  6  m  — 7  x^ 


—  42  ma;3  +  49  x^ 

—  42  mx^  +  49  a;6         =  (6  m  -  7  x^)  (—  7  x^) 


(ii)  The  above  plan  for  extracting  the  square  root  of  a  trinomial  power  is  easily 
extended  so  as  to  apply  to  polynomial  powers  of  any  number  of  terms. 

Consider,  for  example,  the  expression  A  +  k  +  B,  wherein  A  stands  for  the 
first  n  terras,  k  for  the  next  term,  and  B  for  all  the  remaining  terms  of  any  poly- 


*  For  the  consideration  of  the  negative  root  (viz.,  —  A),  see  note  2,  page  211. 


210  ELEMENTARY  ALGEBBA  [Ch.  XIII 

nomial  whatever;  and  let  all  of  the  terms  of  this  polynomial  be  regarded  as 
already  arranged  according  to  the  descending  powers  of  some  one  of  its  letters. 

The  square  of  this  polynomial  is  A^  +  2  Ak-\- k^-{-2  AB -i-2  Bk-{-  B^,  and  the 
question  is:  c/iven  the  power  A^-{-2  Ak  +  k^  +  2  AB -{-2 Bk  + B'^,  how  may  the 
root  A-{-  k  +  B  be  found  from  it  ? 

Let  it  be  assumed  that  the  terms  represented  by  A  have  already  been  found,* 
—  by  (i)  above  or  any  method  whatever,  —  then  it  is  clear  that  when  A^  has  been 
subtracted  from  the  power,  the  highest  term  in  the  remainder  is  the  highest  term 
in  2Ak,  hence  the  next  term  in  the  root  (viz.,  k)  may  be  found  by  dividing  the 
highest  term  in  this  remainder  by  the  highest  term  in  2  A,  i.e.,  by  the  highest 
term  in  the  trial  divisor.  But  since  A  stands  for  the  terms  of  the  root  already 
found,  therefore  what  has  just  been  said  shows  how  to  find  the  next  term  of  the 
root  at  any  stage  of  the  work,  i.e.,  it  shows  how  to  find  all  the  terms  of  the 
root. 

This  work  may  be  arranged  thus : 


A^  +  2Ak  +  k^  +  2AB^2Bk  +  B2     \A+k 
A^ 


Trial  divisor,  2  A 

Complete  divisor,  2A-^k 


2Ak  +  k'^-\-2AB  +  2Bk+B^ 

2Ak  +  k^  =i2A  +  k)'k 


2AB  +  2Bk  +  B^ 


Observe  that  the  two  subtractions  here  made  are  together  equivalent  to  sub- 
tracting (^  +  A;)2from  the  given  power;  i.e.,  by  proceeding  as  above  explained, 
the  remainder  at  any  stage  of  the  work  is  the  same  as  that  obtained  by  subtract- 
ing the  square  of  the  root  found  at  that  stage  of  the  work  from  the  given  power. 

Similarly,  to  find  the  square  root  of  9x^  +  6  xhj  —  11  xhj'^  —  4  xij^  +  4  y^,  the 
work  may  be  arranged  thus  (the  student  should,  however,  explain  each  step) : 

9  a:4+r,.r3,/_iia;2;/2_4  3^2/3+4  ?/4     [  3  x^+a;?/— 2  ?/2 
9a;4 


(>  xhj  - 1 1  a:2?/2— 4  a;?/3+4  ?/4 

6x3//+     x%2  ={Qx'^+xy)  'Xy 


1st  trial  div.,  2{Zx'^)=i\x'^ 

1st  comp.  div. ,  6  x^+xy 

2d  trial  div.,    2(3a;2-f-a-//)=6x2+2x?/|  — 12.r2?/2— iajyS-f^^/^ 

2d  comp.  div. ,  6 a;2+2  xy-2  r/2  j  —12  x^y"^-^ a!?/3+4 ?/4=  (fi  x'^+2 xy—2 y2)  .27/2 


The  above  method  for  extracting  the  square  root  of  a  polynomial 
may  be  stated  thus  : 

(1)  Arrange  the  terms  of  the  given  polynomial  according 
to  the  descending  powers  of  soine  one  of  its  letters,  and 
write  the  square  root  of  its  first  term  as  the  first  term  of 
the  required  root. 

*  The  first  term  at  least  may  always  be  found  as  in  (i)  above. 


125]  INVOLUTION  AND  EVOLUTION  211 

(2)  Subtract  the  square  of  the  root  terrn  just  found  from 
the  given  polynofnial,  and  divide  the  first  term  of  the 
remainder  by  twice  tl%e  first  termj  of  the  root;  write  the 
quotient  as  the  next  terin  of  the  required  root,  and  also 
annex  it  to  the  trial  divisor  to  form  the  complete  divisor. 

Q^)  Multiply  the  coTnplete  divisor  by  the  last  root  term, 
which  has  just  been  found,  and  subtract  the  product  from 
tixe  preceding  remainder. 

(4)  Divide  the  first  term  of  this  new  remainder  by  the 
first  term  of  the  new  trial  divisor;  write  the  quotient  as 
the  next  term  of  the  required  root,  and  also  add  it  to  the 
trial  divisor  to  form  the  complete  divisor. 

(5)  Repeat  the  steps  (3)  and  (4)  until  all  the  terms  of 
the  root  are  found. 

Note  1.  Observe  that  if  polynomials  are  arranged  according  to  ascending 
instead  of  to  descending  powers  of  the  letter  of  arrangement,  the  above  demon- 
stration still  applies;  it  requires  only  the  verbal  change  of  lowest  term  for 
highest  term. 

Note  2.  If  the  negative  value,  instead  of  the  positive  value,  of  the  square 
root  of  the  first  term  of  the  polynomial  had  been  used  in  the  above  demonstra- 
tion, the  sign  of  each  term  of  the  result  would  have  been  changed,  i.e.,  the  result 
would  have  been  the  negative  square  root  of  the  given  polynomial. 

Note  3.  It  has  been  shown  above  how  to  find  the  square  root  of  a  polynomial 
which  is  an  exact  square;  i.e.,  if  the  above  process  be  continued  until  a  zero 
remainder  is  reached,  then  the  square  of  the  expression  thus  found  will  be  the 
given  polynomial.  If,  however,  the  same  process  be  applied  to  a  polynomial 
which  is  not  an  exact  square,  then  as  many  root  terms  as  desired  may  be  found, 
and  the  square  of  this  root,  at  any  stage  of  the  work,  will  equal  the  result  of  sub- 
tracting the  corresponding  remainder  from  the  given  polynomial  — such  a  root  is 
usually  called  an  approximate  root,  and  also  the  root  to  n  terms. 

EXERCISES 

Find  the  square  root  of  each  of  the  following  expressions,  and  verify 
the  correctness  of  your  result : 

1.  a;*  -  4  a;8  +  8  X  +  4. 

2.  4  w*  -  4  m3  -  3  m2  +  2  Tw  +  1. 

3.  l-6y+52/2+i2y3_^.43,4.  . 

4.  25  x^  -  40  a%^x^y^  +  16  a^ft*. 

5.  4x6  +  i7a.2  _  22  x3  +  13  x4  -  24  X  -  4x6  +  16. 


212  ELEMENTARY  ALGEBRA  [Ch.  Xlll 

6.  4  a4  +  64  64  _  20  a%  +  57  a^"^  -  80  ab^ 

7.  6  a:^^  +  2  x^y^  -  28  xy^  -\- 9  x^  +  4:  y^  +  45  a;^?/*  +  43  x^y^. 

8.  3  ^4  -  2  x5  -  a:2  +  2  a:  +  1  +  x^  * 

9.  48  a*  +  12  a2  +  1  -  4  a  -  32  a8  +  64  a«  -  64  a^. 

10.  46  a:2  +  25  a;4  -  44  a:3  -  40  ar  +  4  a:«  +  25  -  12  x^. 

11.  x^-2x'^y  +  2  x^z^  -  2  7/^2  +  y2  ^  2*. 

12.  a:8  -  2  a'^x^  -  3  a^a;*  +  4  «6a;2  +  4  «»  - 16  a^a;+  32  aSa;^- 20  a3a;5  +  4  aa;^ 

14.  —  +  16  a^y^  +  8  x'^y^. 

15.  a;2+2a;-  1  --  +  --t 

X      x^ 

16.  9a,-2-24a;  +  28-  — +  i. 

a;       a:^ 

17.  n4  +  4  n3  +  -  +  2  n  +  4  4-  4  w2, 

18.  x*  +  1  +  4  a;8  +  -  +  6  a:2  +  -^  +  5  +  5  a:  +  -. 

x^  x^  4a;2  a; 

19.  4  +  ^'-^-^  +  -^. 

62         6       a      4a2 

20.  (a:  -  y)2  -  2  (a-^/  +  3:2  -  2/2  _  yz)  +  (y  +  2)2. 

21.  a;2'-_y2»  _  6  x^^hf^"^  -  30  x^y'^"^  +  10  a:2''-y«+i  +  25  a;2'-23/2s+2  _f.  9  ^.y, 

22.  1  +  a:,  to  4  terms.     See  note  3. 

23.  a2  +  1,  to  3  terms. 

24.  1  +  a;  —  a:2,  to  4  terms. 

25.  a;4  +  2  x^y  +  2/^  +  a:^/^  +  a:23^^  to  4  terms. 

26.  By  extracting  the  square  root  until  a  numerical  remainder  is 

reached,  show  that  x4  +  4x^+8a:2  +  8a:-  5  equals  (^2  +  2  x  +  2)2  -  9, 
and  thus  find  the  factors  of  x*  +  4  a;^  +  8  a:2  +  8  x  —  5. 

27.  Similarly,    find    the   factors    of    x*  +  6  x^  +  11  x2  +  6x  -  8    and 
a6  -  6  a*  +  10  a8  +  9  a2  _  30  a  +  9. 

*  Check  Exs.  8-21  by  the  method  of  Ex.  7,  §  39. 

t  Show  first  that  this  expression  is  already  arranged  according  to  descending 
powers  of  «. 


125-126]  INVOLUTION  AND  EVOLUTION  213 

126.  Square  roots  of  arithmetical  numbers.  Arithmetical  num- 
bers are  merely  disguised  polynomials  —  e.g.,  3862  =  3  (10)*  + 
8  (10)^  +  6  (10)  +  2  —  and  their  square  roots  are  extracted  by 
virtually  the  same  process  as  that  given  in  the  preceding  article. 

Although  it  is  not  necessary  to  do  so,  yet  it  is  more  systematic 
to  find  the  several  digits  of  these  roots  in  their  order  from  left  to 
right,  just  as  the  terms  are  found  in  the  case  of  polynomials; 
to  do  this  the  given  number  is  first  separated  into  periods  of  two 
figures  each,  to  the  right  and  left  of  the  decimal  point. 

The  reason  for  the  separation  into  periods  lies  in  this :  the  square  of  any  num- 
ber of  tens  ends  in  two  ciphers,  and  hence  the  first  two  digits  at  the  left  of  the 
decimal  point  are  useless  when  finding  the  tens'  digit  of  the  root ;  they  are  there- 
fore set  aside  until  needed  to  find  the  units'  digit  of  the  root.  So,  too,  the  square 
of  any  number  of  hundreds  ends  in  four  ciphers,  and  hence,  for  a  like  reason, 
two  periods  are  set  aside  when  the  hundreds'  digit  of  the  root  is  being  found,  and 
so  on.    Similarly  for  the  periods  at  the  right  of  the  decimal  point. 

The  application  of  the  method  of  §  125  to  extracting  square 
roots  of  arithmetical  numbers  may  be  best  understood  in  general 
by  first  considering  some  particular  examples. 

Let  it  be  required,  for  instance,  to  find  the  square  root  of  1156. 

Since  this  number  consists  of  two  periods,  therefore  its  square  root  will  consist 
of  two  integer  places,  i.e.,  of  tens  and  units. 

Moreover,  since  30^  <  1156  <  40^,  therefore  the  required  root  lies  between  30 
and  40,  i.e.,  the  tens'  digit  is  3,  the  square  root  of  the  greatest  square  integer  in 
the  left-hand  period  of  the  given  number. 

The  units'  digit  may  now  be  found  as  follows :  let  Tc  represent  the  part  of  the 
root  already  known  (viz.  30),  and  let  u  represent  the  unknown  part  of  the  root ; 

then  1156  =  (A;  +  w)2  =  A:2  +  2  ku  +  w2, 

and,  therefore,  2  ^w  +  w^  =  1156  -  A;2  =  256.  [A;2=900 

Again,  since  k  represents  tens  while  u  represents  units,  therefore  2  ku  is  much 
greater  than  ifi ;  hence  the  last  equation  above  shows  that  2  kit  (though  somewhat 
less  than  256)  is  approximately  equal  to  256,  and  hence  that  256 -^2  A;  (though 
somewhat  too  great)  is  approximately  equal  to  u,  i.e.,  256 -^  2  Z:  will  suggest  a 
value  for  u,  which  must  then  be  tested  by  the  above  equation.* 

*  Since  2.56=  (^k-\-u)u,  therefore  256^  (2  A;4- w)  ==  w,  i.e.,  the  complete  divi- 
sor is  2k-\-u,  and  2fc  is  merely  a  trial  divisor;  hence  the  appropriateness  of 
these  names.  Since  256-^2  ^  gives  too  great  a  quotient,  therefore  the  units'  digit 
in  the  required  square  root  is  either  4  or  a  smaller  number;  hence  if  the  units' 
digit  is  not  4  (i.e.,  if  it  is  3,  2,  1,  or  0),  then  (A:  +  4)2  >  1156,  i.e.,  1156—  (A: +  4)2 
is  negative,  and  the  next  smaller  number  must  be  tried.  This  shows  that  the  Jirst 
one  of  these  numbers  (4,  3,  •••)  which  leaves  a  positive  remainder  in  the  above 
subtraction  is  the  units'  digit  in  \/ll56.    Similarly  in  general. 


214 


ELEMENTARY  ALGEBRA 


[Ch.  XIII 


Finally,  since  k  is  already  known  to  be  30,  therefore  256  -;- 2  A:  =  256 -^  60  =  4+, 
hence  u  is  probably  equal  to  4;  substituting  this  value  of  u  in  the  equation 
266  =  2  A;u  +  w2^  proves  that  w  =  4,  and  hence  that  \/ll56  =  34. 

The  work  may  be  arranged  as  follows : 

(A;  +  w)2=A;2  +  2A:n  +  w2=     11'56      1 30  +  4  =  34 
^2  =  (30)2  =      900 
trial  divisor  is  2  A;  =  60  \25Q  =  2  ku  +  u^ 

complete  divisor  is       2  &  +  m  =  64  1256=  {2k-\-u)-u 

0 

Again,  let  it  be  required  to  find  the  square  root  of  315844. 

Since  this  number  consists  of  three  periods,  therefore  its  square  root  will  con- 
sist of  three  integer  places.  The  work  may  be  arranged  as  follows  (the  student 
should  fully  explain  each  step) : 


31'58'44 
250000 


1500  +  60  +  2  =  562 


1st  trial  divisor,  2  •  500  =  1000 

Ist  complete  divisor,  1000  +  60  =  1060 
2d  trial  divisor,  2  •  560  =  1120 

2d  complete  divisor,    1120  +  2  =  1122 


&5844 

63600  =  1060  .  60 


12244 

1 2244=  1122  .  2 
0 


NoTB.  When  some  familiarity  with  the  above  process  has  been  gained,  the  work 
may  be  abridged  by  omitting  unnecessary  ciphers,  and  annexing  to  each  remainder 
the  two  digits  which  compose  the  next  period  in  the  given  number,  thus : 


31'58'44 
25 


1562 


Ist  complete  divisor,  106 
2d  complete  divisor,  1122 


658 
636 


12244 

12244 

0 


Finally,  let  it  be  required  to  extract  the  square  root  of  10.5626. 

13.25 


The  work  may  be  arranged  thus : 

1st  complete  divisor,  62 
2d  complete  divisor,  646 


10.'56'25 
9 


156 
124 


13225 

13225 

0 


The  results  of  the  discussion  of  the  present  article  may  be  stated 
thus: 

(1)  Separate  the  given  number  into  periods  of  two  digits 
each,  beginning  at  the  decimal  point  and  counting  both 
toward  the^  right  and  toward  the  left,  completing  the  right- 
hand  decimal  period  by  annexing  a  cipher  if  necessary. 


126]  INVOLUTION  AND  EVOLUTION  215 

(2)  By  inspection  find  the  greatest  square  integer  in  the 
left-hand  period,  and  write  its  square  root  as  the  first  digit 
of  the  required  root. 

(3)  Subtract  the  square  of  the  root  digit  already  found 
from  the  left-hand  period  of  the  given  number,  and  bring 
down  the  next  period  as  part  of  the  remainder. 

(4)  Divide  this  remainder,  exclusive  of  its  right-lxand 
digit,  by  twixie  the  root  digit  already  found,  i.e.,  by  tJie 
tidal  divisor,  and  annex  tlxe  quotient  digit  to  the  root 
and  also  to  the  trial  divisor,  thus  forming  the  complete 
divisor. 

(5)  Multiply  the  complete  divisor  by  the  last  digit  in  the 
root,  subtraxit  the  product  from  the  former  remainder, 
and  bring  down  the  next  period  of  the  given  nuinber  as 
part  of  this  new  remainder. 

(6)  Repeat  (4)  and  (5)  above  until  all  the  periods  of 
the  given  number  are  exhausted. 

(7).  //  a  negative  remainder  presents  itself  in  the  above 
worh,  it  indicates  that  the  corresponding  trial  root  digit  is 
too  great,  and  the  one  next  lower  must  be  tried. 

(8)  For  a  given  number  which  is  not  a  perfect  square 
as  many  decimal  figures  as  desired  in  the  root  may  be 
found  by  annexing  the  necessary  number  of  periods  of 
ciphers  to  the  number  (cf.  §  125,  note  3). 

EXERCISES 

Extract  the  square  root  of  each  of  the  following  numbers  : 

1.  1296.  3.   7396.  5.    667489.  7.   17424. 

2.  841.  4.   12.96.  6.    1664.64.  8.    101.0025. 

9.   How  may  the  square  root  of  a  fraction  be  found?     Why?     What 

is  the  square  root  of  -^-^  ?     Why  ? 

10.  Find  the  square  root  of  f f|.  Is  —  |f  also  a  square  root  of  this 
fraction?    Why? 

11.  If  a  number  contains  3  decimal  places,  how  many  decimal  places 
does  the  square  of  this  number  contain  ?     Why  ?     Generalize  this  relation. 

12.  Extract  the  square  root  of  2  to  three  decimal  places.  How  many 
decimal  ciphers  must  be  annexed  to  2  for  this  purpose  ?     Why  ? 


216  ELEMENTARY  ALGEBRA  [Ch.  XIII 

Find  the  square  root  of  each  of  the  following  numbers,  correct  to 
three  decimal  places : 

13.    13.5.  14.   .017.  15.   |.  16.   4f. 

17.  Show  by  actual  trial  that,  having  found  the  square  root  of  35.8 
correct  to  3  decimal  places,  the  next  2  decimal  figures  of  the  root  may  be 
found  by  simply  dividing  the  remainder  at  that  stage  of  the  work  by  the 
corresponding  trial  divisor. 

18.  If  the  square  root  of  a  number  is  desired,  correct  to  2  n  +  1  figures, 
prove  that  when  the  first  n  +  1  figures  have  been  found  in  the  usual 
way,  the  remaining  n  figures  may  be  found  by  ordinary  division 
(cf.  Ex.  17). 

Suggestion.  Let  N  stand  for  any  number  whatever,  k  for  the  first  n+1 
figures  of  its  square  root  (with  n  ciphers  annexed),  and  r  for  the  remaining  n 
figures  of  the  root. 

Then  N  =  {k+ r)'^  =  k'^  +  2kr+r^, 

whence         — ~  =  r-\ ,     in  which  -^—  is  a  proper  fraction  (why  ?) ; 

2k  2k  2k 

i.e.,  merely  dividing  N—  k^  (which  is  the  remainder  when  the  first  n  + 1  figures 
have  been  found)  by  the  trial  divisor  at  that  stage  of  the  work  (viz.,  2  k)  gives  the 
next  n  figures  of  the  root,  together  with  a  proper  fraction. 


19.  Find  \/84256  to  5  figures,  V3.642  to  3  figures,  and  \/6018274 
to  3  decimal  places.  How  many  root  figures  must  be  found  by  the 
usual  process,  in  each  of  these  cases,  before  the  ordinary  division  may 
begin  ? 

127.  Cube  root  of  polynomials.  The  general  method  for  extract- 
ing the  square  root  of  a  polynomial,  which  is  given  in  §  125, 
may  easily  be  extended  so  as  to  apply  to  cube  root  also  —  and 
indeed  to  the  higher  roots  as  well.  The  process  is  in  all  cases 
the  inverse  of  that  employed  in  raising  a  polynomial  to  a  power. 
The  several  steps  are  indicated  below.* 

Since  (k  +  uy=J^  +  S  k'n  +  3  hi'  +  u^  (1) 

=  k'  +  (3k'-{-3ku-{- u')u,  .  (2) 

therefore : 


*  To  avoid  needless  repetition  here  the  student  is  referred  for  fuller  statement 
of  reasons  to  the  detailed  explanation  already  given  in  §  126. 


126-127]  INVOLUTION  AND  EVOLUTION  217 

(1)  Ajrange  the  terms  of  the  given  polynomial  according 
to  the  descending  powers  of  some  one  of  its  letters. 

(2)  The  highest  term  of  the  required  root  is  the  cube  root 
of  the  highest  teriTi  of  the  given  power ;  i.e.,  the  highest 
term  in  the  above  root  is  VA;^  viz.,  k. 

(3)  If  the  cube  of  the  part  of  the  root  already  found  be 
subtracted  from  the  given  polynoinial,  the  remainder  will 
be  3  Tihi  +  3  kii?'  +  u^,  and  the  next  term  of  the  root  may  be 
found  by  dividing  the  first  term  of  this  remainder  by 
three  tiines  the  square  of  the  first  term  of  the  root  {which 
is  already  known);  i.e.,  t1%e  second  term  of  the  root  is 
3  k'^u  -J-  3  k^,  viz.,  u. 

The  trial  divisor  here  is  3  •  Tc^,  i.e.,  it  is  three  tiines  the  square  of 
the  root  already  known ;  and,  from  Eq.  (2)  above,  it  is  clear  that 
the  complete  divisor  is  3  k^  -}-3kii  -\-  u^,  i.e.,  it  is  the  trial  divisor, 
plus  three  times  the  product  of  the  last  term  of  the  root  by  the 
preceding  part  of  the  root,  plus  the  square  of  the  last  term  of  the 
root. 

The  work  may  be  put  in  the  following  form : 

k^+Zk^u-\-Zku^+u^\k-\-u 
;fc8  ' 

Trial  divisor,  3  •  k^ 


Complete  divisor,       Zk'^-\-Zku-\-u^ 


3A;2M  +  3ytw2  +  tf3 

^k^u-\-Zku'^+u^  =  {Zk'^  +  Zku  +  u^)  'U 


Observe  that  the  two  subtractions  just  performed  are  together  equivalent  to 
the  subtraction  of  {k  +  w)^  from  the  given  polynomial. 

(4)  By  proceeding  as  in  §  125  (ii)  it  is  easy  to  show  that, 
having  found  any  jiumber  of  terms  of  the  required  root, 
and  having  subtracted  tl%e  cube  of  this  part  of  the  root 
from  the  given  polynomial,  the  next  root  term  may  be 
found  by  dividing  the  first  term  of  the  remainder  by  the 
first  term  of  the  trial  divisor,— the  trial  divisor  being 
three  times  the  square  of  the  part  of  the  root  already  found. 
By  continuing  this  process  all  the  terms  of  the  required 
root  may  be  found. 


218 


ELEMENTARY  ALGEBRA 


[Ch.  Xlll 


The  work  of  finding  the  cube  root  of  x^  -  9  x^  +  30  a;4  _  45  a;8  +  30  a;2  _  9  ^  +  1 
may  be  arranged  as  follows : 

x6-9x5  +  30x4-45a;8  +  30a;2_9a;  +  i|;c2_3j;-}-i 

(X2)8=:x6 


9  a;S  +  ;«x4 -45x8  + 30x2-9x4-1 
9x5  +  27x4-27x3 


3x4-18  x3  +  30x2_9a;  +  i 
3x4— 18x3  + 30x2- 9x  +  l 


1st  trial  divisor, 

3(X2)2  =  3X4 

Ist  complete  divisor, 

3x4-9x3  +  9x2 
2d  trial  divisor, 

3(x2-3x)2=  3x4- 18x3  +  27x2 
2d  complete  divisor, 

3x4-18x3  +  27x2 

3x2-9x  +  l' 

3x4_i8x8+-30x2-9x  +  l 

The  student  may  now  solve  this  example  by  arranging  the 
terms  according  to  the  ascending  powers  of  x  and  compare  his 
result  with  the  above. 

EXERCISES 

Find  the  cube  root  of  each  of  the  following  expressions,  and  verify 
the  correctness  of  your  results : 

1.  8x3-12a;2  +  6x-l. 

2.  27  x^  -  189  x^y  +  Ul  xi/ -  S^S  y». 

3.  125n8-150mn2-8m3  +  60  7w2/,. 

4.  675  M2y  +  1215  wy2  +  125  m8  +  729  y3. 

5.  a;«-20a;3-6a:  +  15x4-6a:5+15a:2+l. 

6.  3  a;5  +  9  x^  +  x6  +  8  +  12  a:  +  13  a:3  +  18  z2. 

7.  342  a;2  _  108  a;  -  109  x^  +  216  +  171  x*  -  27  x^  +  27  x^ 

8.  156  x4  -  144  x^  -  99  x3  +  64  x^  +  39  x^  -  9  x  +  1. 


48  ,  108 


12  x2.* 


10.   20 


15 


+  15  c2  +  c«  +  -^  +  c-6  +  6  c\ 


11.  30  2/-1  +  8  2/-3  +  8  2/8  +  30  y  -  12  ?/2  _  25  -  12  y-^. 

12.  6  a^x*  -  4  a3x6  -  2  a^x^  +  6  a^x"^  +  3  a^x  +  a^  +  x^  -  3  ax^. 

13.  108  3/62  -27  y^-  90  2j^z^  +  8  z^  _  80  yh^  +  60  y^z^  +  48  yz^ 


*  Compare  §  125,  Ex.  15. 


127-128]  INVOLUTION  AND  EVOLUTION  219 

14.  ^^i^^8fa/>3^M,2_«6(,,2/,2  +  o)rH3fa36+«V4  +  ^'_35»c. 

h^         b  \  aJ  \  hi        a^        a 

15.  x^  +  a%^  -  3  a%^x  -  3  ahx^  +  3  a^>  ( 1  +  ah)x^  +  3  a%'^{l  +  ah)x'^ 
-  a262(6  4-  aJ).r3. 

16.  a:3y-3  +  x-Y  +  3  x}j-\{y-^  -  1)  +  3  x-hj{x--  -  1)  +  3  x-^y-\l  +  a:-2 
+  y~'^)  -  3  a:''^-^  -  3  a-V  -  xV  +  x-^y-^  +  3  xy  (^2  j^  y^  -  \). 

17.  64  y3n_{.117  y3»-3^10  y3H-2_G  ^,3n-4_36  y3n-5_144  y3n-l  _  g  y3n-6, 

18.  Find  the  first  3  terms  of   V\  +  x. 


19.   Find  the  first  4  terms  of    v  1  -3x4-  x\ 

128.  Cube  root  of  arithmetical  numbers.  To  extract  the  cube 
root  of  an  arithmetical  number,  proceed  as  follows :  * 

(1)  Separate  tJie  given  number  into  periods  of  three  digits 
each,  beginning  at  the  deciinal  point  and  counting  both 
toward  the  right  and  toward  the  left,  completing  the  right- 
l%aihd  deciinal  period  by  annexing  one  or  two  ciphers  if 
necessary. 

(2)  By  inspection  {or  by  trial)  find  the  greatest  cube 
integer  in  the  left-lxaiul  period,  and  write  its  cube  root  as 
the  first  digit  of  the  required  root. 

(3)  Subtract  tl%e  cube  of  the  root  digit  just  found  from 
the  left-hand  period  of  the  given  number,  and  bring  down 
tlie  next  period  as  part  of  the  remainder. 

(4)  To  three  times  the  square  of  the  root  digit  already 
found  annex  two  ciphers,  thus  forming  tlxe  trial  divisor; 
divide  the  above  remainder  by  this  trial  divisor,  and  annex 
tl%e  first  quotient  digit  to  the  root. 

(5)  To  the  trial  divisor  add  three  times  the  product  of 
the  last  root  digit  multiplied  by  the  part  of  the  root  previ- 
ously* found  with  a  cipher  annexed,  and  also  the  square 
of  the  last  root  digit,  thus  forming  the  complete  divisor. 
Multiply  the  complete  divisor  by  the  last  root  digit,  and 
subtract  the  product  from  the  above  remainder,  bringing 
down  the  next  period  as  part  of  the  new  remainder. 


*  The  reasoning  here  is  similar  to  that  given  in  §  126,  and  should  be  given  by  the 
student. 


220 


ELEMENTARY  ALGEBRA 


[Ch.  XIII 


(6)  Repeat  (4)  and  (5)  above  until  all  tl%e  periods  of  the 
given  number  are  exhausted. 

Note.  As  in  the  case  of  square  root  (§  126),  so  here,  if  a  negative  remainder 
presents  itself  in  the  course  of  the  above  worlc,  it  indicates  that  the  correspond- 
ing trial  root  digit  is  too  great,  and  tlie  next  lower  digit  must  be  tried. 

As  many  decimal  figures  as  desired  in  the  root  may  be  obtained  by  annexing 
the  necessary  number  of  periods  of  ciphers  to  a  number  which  is  not  a  perfect 
cube. 

The  work  of  finding  the  cube  root  of  9800344  may  be  arranged  as  follows : 


9'800'344    1214 
8 

1800 

[1800-1200 

1261        =  1261  •  1 

539^ 

[539344 -f- 132300 

539M4=  134836 -4 

1st  trial  divisor,  1200 
1st  correction,  60 

2d  correction,  1 

1st  complete  divisor,  1261 

2d  trial  divisor,  132300  5393M  [539344^-132300^=4  + 

1st  correction,  2520 

2d  correction,  16 

2d  complete  divisor,  134836 


Verification  of  the  correctness  of  the  above  root :  (214)3  —  9800344. 
Again,  let  it  be  required  to  find  the  cube  root  of  43614208. 


Trial  divisor, 
1st  correction, 
2d  correction, 
Complete  divisor. 


43'614'208    136 

27 


2700 

540 

36 

3276 


16614 


19656 


[16614  +  2700  =  6+ 


Since  the  remainder  would  be  negative,  therefore  the  trial  digit  6  is  too  great, 
and  5  must  be  tried. 

43'614'208    1 352 

27 


1st  trial  divisor,  2700 

1st  correction,  450 

2d  correction,  25 

1st  complete  divisor,  3175 

2d  trial  divisor,  367500 

1st  correction,  2100 

2d  correction,  4 


16614 


15875       =3175-5 


739208  [739208  +  367500  =  2  + 

739208  =  369604  •  2 


2d  complete  divisor,     369604 


Verification  of  the  correctness  of  this  root :  (352)3  =  43614208. 


128-129]  INVOLUTION  AND  EVOLUTION  221 

EXERCISES 

Extract  the  cube  root  of  each  of  the  following  numbers : 

1.  1728.  3.   31855.013.  5.   39304. 

2.  571787.  4.   148877.  6.  426.957777. 

7.  305.909539272.  9.   .04,  to  3  decimal  places. 

8.  34.7,  to  2  decimal  places.  10.   3|,  to  2  decimal  places. 

11.  If  the  cube  root  of  a  number  consists  of  2  n  -{■  2  figures,  show- 
that  when  n  +  2  of  these  figures  have  been  obtained  by  the  ordinary 
method,  the  ramaining  n  figures  may  then  be  found  by  simple  division 
(cf.  Ex.  18,  §  126). 


12.  By  the  method  of  Ex.  11,  find  V.0783259  correct  to  6  decimal 
figures. 

129.  Higher  roots  of  polynomials  and  of  numbers.  The  methods 
for  extracting  the  square  and  cube  roots  of  polynomials  which 
are  given  in  §§  125  and  127,  respectively,  may  be  easily  extended 
so  as  to  apply  to  the  higher  roots. 

E.g.,  the  identity  (k  +  u)^  =  k^ -{- 4:  k^u  +  6  k'^u'^  +  ^  ku^  +  n^  shows  that  the 
first  term  of  the  fourth  root  is  the  fourth  root  of  the  Jirst  term  of  the  power,  i.e., 
of  the  given  polynomial;  again,  if  k  and  u  represent  respectively  the  known  and 
unknown  parts  of  the  root  at  any  stage  of  the  work,  and  if  k^  be  subtracted  from 
the  power,  the  remainder  may  be  written  thus :  (4  A;^  +  G  k'^a  +  4  kv!^  +  u^)  a,  which 
shows  that  the  trial  divisor  is  4  k^,  and  that  there  are  three  corrections,  viz.,  6  khi, 
4  ku^,  and  %fi,  which  must  be  added  to  the  trial  divisor  to  give  the  complete  divisor. 
From  here  on  the  work  proceeds  as  in  the  case  of  cube  root. 

Similarly,  in  extracting  the  fifth  root  the  trial  divisor  is  5k^,  and  there  are 
four  corrections  to  be  added  to  the  trial  divisor  to  form  the  complete  divisor;  in 
the  nth  root  (where  n  is  any  positive  integer)  the  trial  divisor  is  nk"--^,  and  there 
are  n  —  1  corrections. 

The  method  of  extracting  any  root  of  a  polynomial  is  easily  adapted  to  the 
extraction  of  the  corresponding  root  of  an  arithmetical  number,  as  has  already 
been  illustrated  in  §§  12G  and  128. 

Note.  If  a  number  be  separated  into  two  equal  factors,  and  each  of  these  two 
factors  be  further  separated  into  three  equal  factors,  the  given  number  will  then 
really  have  been  separated  into  6  (i.e.,  3 '2)  equal  factors;  from  this  it  follows 
that  if  N  represents  a  number  which  can  be  separated  into  6  equal  factors,  then 
</^  =  </^. 

Similarly,  in  general,  if  N  represents  a  number  which  can  be  separated  into 
p  '  q  equal  factors,  then  'Vn  =  V  Vn  =  "V  VN.  This  fact  simplifies  the  extrac- 
tion of  the  higher  roots  whenever  the  index  of  the  required  root  is  a  composite 
number  (cf.  also  §  136). 


222  ELEMENTARY  ALGEBliA  [Ch.  XIII 

EXERCISES 

Find  the  indicated  roots  of  the  following  expressions  —  both  directly, 
and  also  by  the  method  given  in  the  preceding  note : 

1.  ^/x^  -  8  ;r8  +  24  a;2  ~  32  X  +  16. 

2.  \/81  /  +  54  xhf^  +  a:4  +  12  x^y  +  108  xy^. 

3.  Vx^  -  12  x^  +  60  a;4  -  160  x^  +  240  x^  -  192  x  +  64. 

4.  \/15  a^c^a;^  +  ae^e  +  ^  ^5^53.  _^  20  a^c^x^  +  15  a%2^4  +  a-e  +  6  acxK 

5.  Find  the  fifth  root  of  32  a:^  4.  80  x^  +  80  x^  +  40  a;2  +*10  a;  +  1. 

Find  the  following  indicated  roots: 

6.  </v}^ + 243  yio  + 1 5  M8y2  +  405  uH^ +90  u^v^  +  270  uH^. 

7.  v/50625.  8.    \/53r44l.  9.    \/5764801.  10.    \/1874161. 


CHAPTER   XIV 

IRRATIONAL  AND   IMAGINARY  NUMBERS  —  FRACTIONAL 
EXPONENTS 

I.   IRRATIONAL   NUMBERS 

130.  Preliminary  considerations  and  definitions.  While  such 
roots  as  V4,  V— /y,  V32aV",  etc.,  can  be  exactly  expressed  by 
means  of  integers  and  fractions,  many  others  which  frequently 
present  themselves  in  algebraic  investigations  can  not  be  so 
represented;  e.g.,  V2  and  V—  5. 

These  new  numbers,  and  their  laws  of  combination,  will  now  be 
examined,  and  they  will  henceforth  be  included  in  the  number 
system,  which  heretofore  has  comprised  only  positive  and  nega- 
tive integers  and  fractions. 

Note  1.  That  \/2  is  neither  an  integer  nor  a  fraction  may  be  shown  as  follows : 
By  the  definition  of  a  root  (§  122),  V2  means  a  number  whose  square  is  2,  and 
since  (+  1)2  <  2  and  (+  2)2  >  2,  therefore  the  number  whose  square  is  2  must,  in 
absolute  value,  lie  between  1  and  2,  and  therefore  can  not  be  an  integer.  Moreover, 
\/2  can  not  be  a  fraction  such  as  —  because  if  it  were,  then  —  would  equal  2,  but 

m  ^  ^*  rrfi 

if  —  is  a  fraction,  it  may  be  supposed  to  be  in  its  lowest  terms,  and  then  — -  is 

n  n^ 

also  a  fraction  in  its  lowest  terms  and  can  not  be  equal  to  the  integer  2.  It  is  then 
proved  that  V2  is  neither  an  integer  nor  a  fraction. 

Note  2.  Although,  as  has  just  been  shown,  such  numbers  as  \/2  can  not  be 
exactly  represented  by  integers  or  by  fractions,  yet  they  can  be  approximately 
represented,  and  to  any  required  degree  of  accuracy,  by  means  of  these  numbers. 

E.g.,  squaring  1,  2,  3,  •••  in  turn  shows  that  1  <  a/2 < 2,  then  squaring  1.1, 
1.2,  1.3,  •..  in  turn  shows  that  1.4  <  \/2<1.5,  then  squaring  1.41,  1.42,  1.43,  •••  in 
turn  shows  that  1.41  <  \/2  <  1.42,  etc. 

Thus  it  is  shown  that  1  <  \/2  <  2,  1.4<V2<1.5,  1.41  <  V2<  1.42, 
1.414  <  v'2  <  1.415,  etc. ;  and  since  a  number  which  lies  between  two  other  num- 
bers differs  from  either  of  them  by  less  than  they  differ  from  each  other,  therefore 
y/2,  differs  from  1  or  2  by  less  than  1,  from  1.4  or  1.5  by  less  than  0.1,  from  1.41 
or  1.42  by  less  than  0.01,  etc.  If,  then,  the  numbers  1,  1.4,  1.41,  1.414,  •••  be  taken 
as  successive  approximations  to  the  value  of  V'2,  the  errors  will  be  less  than  1, 
0.1,  0.01,  0.001,  •••  respectively  ;  hence  it  is  clear  that,  by  continuing  the  above 
process,  a  number  can  be  found  which  can  be  expressed  by  means  of  integers, 
and  which  will  represent  \/2  to  any  required  degree  of  accuracy. 

223 


224  ELEMENTARY  ALGEBBA  [Ch.  XIV 

Furthermore,  it  is  evident  from  the  nature  of  the  argument  just  given  that  it- 
applies  equally  well  to  any  indicated  root  of  a  positive  number,  and  also  to  odd 
roots  of  negative  numbers. 

Note  3.  Although  such  numbers  as  V'2  can  not  be  exactly  expressed  by 
means  of  integers  and  fractions,  they  are  just  as  definite  and  precise  as  are 
integers  and  fractions,  and  they  are  also  necessary  in  human  affairs. 

E.g.,  let  the  figure  ABCD  be  a  square  whose 

Bi ^p  side  AB  is  1  foot  long,  and  let  the  figure 

\  ACEF  be  another  square  whose  side  ^C  is  the 

\  diagonal  of  the  first  square ;  then  it  is  easily 

^\  proved  by  geometry  that  the  area  of  the  square 

ACEF  is  2  times  that  of  ABCD,  and  hence,  if 

X  is  the  number  of  feet  in  AC,  then  a;^  =  2 ;  i.e.. 


^,  BJ 

i  y         if  the  length  of  the  side  of  a  square  is  1  foot, 

\  ;  y  then  the  length  of  the  diagonal  of  that  square 

\  j  /  is  precisely  \/2  feet. 

\      I      /^  This  illustration  shows  also  that  such  num- 

\  I  y  bers  are  necessary  in  human  affairs,  e.g.,  \/2 

m  '  is  the  only  number  which  exactly  expresses 

the  length  of  the  diagonal  of  a  unit  square,  — 

the  numbers  1,  1.4,  1.41,  1.414,  1.4142,  •••  are  successive  approximations  to  the 

length  of  this  diagonal,  but  its  exact  length  is  a  number  whose  square  is  exactly  2, 

and  which  is  represented  by  the  symbol  •v/2; 

Note  4.  That  the  other  root  indicated  above,  viz.,  V—  5,  can  not  be  expressed, 
even  approximately,  by  means  of  integers  and  fractions  follows  directly  from  the 
law  of  signs  in  multiplication ;  if  it  could  be  so  expressed  it  must  be  either  a  posi- 
tive or  a  negative  number,  and  its  square  would  then  be  a  positive  number  and 
not  —  5.  The  same  argument  applies  to  every  indicated  even  root  of  a  negative 
number. 

Numbers  that  involve  indicated  roots  which  can  not  be  exactly 
expressed  by  means  of  integers  and  fractions,  but  which  may  be 
expressed  to  any  required  degree  of  accuracy  by  means  of  these 
numbers,  are  called  irrational  numbers,  while  integers  and  fractions 
are  classed  together  as  rational  numbers. 

E.g.,  yj'l,  4—  v7,  and  V2  +  v5  are  irrational  numbers. 

Numbers  which  involve  indicated  even  roots  of  negative  mwyh- 
bers  are  called  imaginary  numbers,*  and  all  other  numbers  are,  for 
distinction,  called  real  numbers. 

E.g.,  V—  3,  2  +  V--5,  and  3\/—  2  are  imaginary  numbers. 

*  The  name  "  imaginary  "  is  rather  an  unhappy  one  because  these  numbers  are 
just  as  real,  under  their  proper  interpretation,  as  any  other  numbers. 

For  present  jmrposes  it  seems  best  to  define  irrational  and  imaginary  numbers 
as  above,  and  thus  to  separate  them  ;  the  name  "  irrational"  is,  however,  often 
employed  to  include  the  imaginary  numbers  also. 

For  a  broader  definition  of  imaginary  numbers  see  Appendix  B. 


130]  IRRATIONAL  NUMBERS  225 

Although  the  language  employed  in  defining  a  root  of  a  number 
in  §  122  is  general,  and  includes  the  irrational  and  imaginary  roots 
as  well  as  the  rational  roots,  yet  the  student's  conception  of  a  root 
has  doubtless  heretofore  been  limited  to  those  roots  which 
happened  to  be  rational ;  it  is  therefore  worth  while  especially  to 
emphasize  here  that  the  symbol  -{/a  stands  for  a  number  whose 

nth  power  is  a, 

(ya)":  =  a* 

where  a  is  any  number  whatever,  and  the  only  limitation  upon 
the  symbol  is  that  n  must  be  a  positive  integer. 

Note  5.  Having  uow  further  enlarged  the  number  concept,  it  may  be  worth 
while  to  recapitulate  briefly  what  has  already  been  said  upon  this  subject  in  the 
preceding  pages. 

The  first  numbers  which  man  invented  to  express  the  relations  of  the  things 
about  him  were  the  positive  Jntetrersj  with  these  he  found  it  necessary  to  perform 
certain  fundamental  operations  (addition,  subtraction,  etc.),  and  later  he  found 
it  necessary  to  enlarge  his  idea  of  number  so  as  to  make  these  operations  always 
possible  (cf.  §  12,  note).  Thus  fractions  arose  from  generalizing  the  operation  of 
division  (cf.  §  11) ;  negative  numbers  arose  from  generalizing  the  operation  of 
subtraction  (cf.  §§  12-14) ;  and  in  the  present  article  it  appears  that  generalizing 
the  operation  of  extracting  roots  introduces  two  further  new  kinds  of  numbers, 
viz.,  the  irrational  and  the  imaginary. 

In  other  words:  while  the  direct  operations  (viz.,  addition,  multiplication,  and 
involution)  with  positive  integers  always  produce  results  that  are  positive  integers, 
the  inverse  operations  (viz.,  subtraction,  division,  and  evolution)  lead  respectively 
to  negative,  fractional,  and  irrational  and  imaginary  numbers,  and  demand  for 
their  accommodation  that  the  primitive  idea  of  number  be  so  enlarged  as  to  include 
these  new  kinds  of  numbers  along  with  the  positive  integers. 


EXERCISES 

1.  What  is  an  irrational  number?  Show  that  V—  5  is  not  an  irra- 
tional number.     To  what  class  of  numbers  does  V  —  5  belong  ? 

2.  Is  \/8  an  irrational  number?  Why?  Show  that  V5  is  neither  an 
integer  nor  a  fraction.  To  what  class  of  numbers  does  V5  belong? 
Why? 

3.  Find  three  successive  approximations  to  the  value  of  V5  (cf.  note  2 
above).  Compare  these  approximations  with  the  result  of  extracting 
the  square  root  of  5  by  the  method  of  §  126. 

*  It  may  be  remarked  that,  under  this  definition,  Va  means  the  same  as  a  [cf. 
§  7  (iv)  note]. 


226  ELEMENTARY  ALGEBRA  [Ch.  XIV 

4.  Find  two  approximate  values  of  V3,  one  larger  and  the  other 
smaller  than  the  true  value,  which  differ  from  Vs  by  less  than  .001. 

5.  A  fruit  grower  has  16  plum  trees  and  wishes  to  plant  them  in 
rows  in  a  rectangular  plot  of  ground,  and  to  have  the  number  of  trees  in 
each  row  exceed  the  number  of  rows  by  2.  How  many  trees  shall  he 
plant  in  a  row  ? 

Suggestion.  If  x  represents  the  number  of  trees  to  be  planted  in  a  row, 
show  that  z^  —  2z  =  l(i.  From  this  equation  it  follows  tha^t  (a;  — 1)2  — 17  =  0, 
i.e.,  that  (x  - 1  +  VlT) {x-1-  Vl7)  =  0 ;  whence  a;  =  1  +  Vl7,  or  a;  =  1  -  Vl7. 

Does  the  fact  that  one  can  not  plant  1  +  >/l7  trees  in  a  row  show  that 
there  is  no  such  number  as  1  +  VlT?  Or  does  it  merely  show  that  the 
present  problem  demands  what  is  impossible  ? 

6.  Show  how  to  construct  a  line  which  shall  be  exactly  1  +  Vl7  times 
as  long  as  a  given  line. 

7.  Can  V—  8  be  expressed  by  means  of  an  integer  or  a  fraction  ?  Is 
it  then  an  irrational  number?    Why  not?    What  kind  of  number  is  it? 

8.  Is  the  number  21  +  VIZ  rational  or  irrational?  Why  ?  What  kind 
of  number  is  84  V5  -  v/  -^  ?    Why  ? 


131.  Further  definitions.  An  indicated  root  of  a  number  is 
usually  called  a  radical ;  if  this  root  is  irrational,  but  the  radicand 
rational,  the  expression  is  also  called  a  surd. 


E.g.,  V2,  v^8,  >/5  +  \/10,  and6v^45  are  radicals;  and  of  these  \/2and6v^ 
alone  are  called  surds. 

The  coefficient  of  a  radical  is  the  factor  which  multiplies  it,  and 
the  order  of  the  radical  is  determined  by  the  root  index.  Two  radi- 
cals which  have  the  same  root  index  are  said  to  be  of  the  same 
order. 

E.g.,  the  surds  12 y/5  ax^  and  m^V&fi  are  of  the  same  order,  viz.,  the  7th,  and 
their  coeflacients  are  12  and  m^,  respectively. 

Surds  of  the  second  and  third  orders  are  usually  called  quadratic  and  cubic 
surds,  respectively. 

If  two  or  more  radicals  are  of  the  same  order,  and  have  their 
radicands  (cf.  §  122)  exactly  alike  —  or  if  they  can  be  reduced  to 
such — they  are  called  similar  radicals  and  also  like  radicals;  other- 
wise the  J  are  dissimilar  (unlike). 


130-132]  IRRATIONAL   NUMBERS  227 

Expressions  which  involve  radicals,  in  any  way  whatever,  are 
called  radical  expressions ;  they  are  monomial,  binomial,  etc.  (cf. 
§  27),  depending  upon  the  number  of  their  terms. 

E.g.,  Vs  and  3\/5  are  similar  quadratic  surds,  while  6-\/a2  +  2  6a; +  ?/*  and 
{m  +  2.n)y/a^  +  2hx  +  y  are  similar  cubic  surds.  The  four  examples  just  given 
are  monomial  surds,  while  5  a  +  3 V7  and  2^9  +  SVx  are  binomial  surds. 

132.  Principal  roots.  It  has  already  appeared  that  a  number 
has  tioo  square  roots  {e.g.,  V9  is  +  3  or  —  3),  and  it  will  be  seen 
later  that  every  number  has  three  cube  roots,  four  fourth  roots, 
Jive  fifth  roots,  etc. 

E.g.,  ■\/8  =  2,  —  l  +  V— 3,  or  —1  —  ^—3,  since  the  cube  of  each  of  these 
numbers  is  8  (cf.  Ex.  23,  §  170) ;  and   v/I(3  =  2,  -  2,  2  V-  1,  or  -  2\/^. 

Although,  as  has  just  been  said,  a  numbeu  has  3  cube  roots, 
4  fourth  roots,  etc.,  some  of  these  roots  are  imaginary,  and  when 
there  are  two  real  roots,  they  are  equal  in  absolute  value  and  of 
opposite  sign.f 

By  the  principal  root  of  a  number  is  meant  its  real  root,  if  there 
is  but  one  real  root,  and  its  real  positive  root  if  there  are  two  real 
roots. 

E.g.,  if  attention  is  confined  to  principal  roots,  V9=3  (and  not  —3), 
v^-  8  =  -  2,   \^l25  =  5,  ^16  =  2,  etc. 

That  irrational  and  imaginary  numbers  obey  the  fundamental 
combinatory  laws  (commutative,  associative,  etc.)  which  have 
already  been  established  in  the  case  of  rational  numbers  is 
proved  in  the  appendix ;  logically  this  proof  for  irrational  num- 
bers should  now  be  read,  but  it  may  be  deferred  until  later  if  the 
reader  will  carefully  bear  in  mind  that  the  following  discussion 
assumes  that  irrational  numbers  are  subject  to  these  laws,  and 
that  the  results  are  therefore  to  be  regarded  as  tentative  until  this 
fact  is  proved. 


*  Such  expressions  are  said  to  be  surd  in  form  even  though  values  may  be 
assigned  to  the  letters  involved  which  make  them  rational  in  value. 

t  It  should  be  especially  observed  that  a  number  can  not  have  two  real  roots  of 
unequal  absolute  value.  For  suppose  Va  =  r^  and  also  r^,  where  r^  and  r^  are 
real,  and  r^^r^i  in  absolute  value;  from  this  it  follows  that  r^^rc^  in  abso- 
lute value,  and  therefore,  if  rji  =  a,  then  rg**  ^  a,  i.e.,  Va  ^  r^- 


228  ELEMENTARY  ALGEBRA  [Ch.  XIV 

EXERCISES 

1.  What  is  a  radical  expression?  A  surd?  Give  examples  to  illus- 
trate your  answer.     Are  all  radicals  surds?     Are  all  surds  radicals  ? 

2.  What  is  the  coefficient  of  a  surd  ?  Give  an  example.  May  this 
coefficient  be  a  negative  number?  May  it  be  a  fraction?  Are  there 
any  restrictions  upon  it  ? 

3.  What  is  meant  by  the  order  of  a  surd?  Illustrate  by  examples. 
May  the  order  of  a  surd  as  now  defined  be  negative  or  fractional  ? 

4.  Define  similar  surds,  and  illustrate  your  definition  by  several 
examples.     May  the  coefficients  differ  and  the  surds  still  be  similar? 

5.  What  factor  have  any  two  similar  surds  necessarily  in  common? 
What  kind  of  number,  then,  is  the  quotient  of  two  similar  surds  ?  Illus- 
trate your  answer. 

6.  What  is  an  imaginary  number?  Give  several  illustrations.  For 
what  vaUies  of  n  is  "v^— 5  an  imaginary  number?  Give  a  reason  for 
calling  these  numbers  "  imaginary." 

7.  Illustrate  by  examples:  monomial  and  trinomial  surds;  quadratic 
and  cubic  surds ;  and  the  order  of  a  surd. 

8.  How  many  values  has  Vl6?  What  are  they?  What  is  the 
principal  square  root  of  16?  What  is  the  principal  fifth  root  of  —  32? 
Define  the  principal  root  of  a  number. 

9.  Show  that  \/'d4:3  is  7.  Under  what  conditions  is  VK  equal  to  ;?? 
How,  in  general,  is  the  correctness  of  a  root  tested  ? 

10.   Show  that  under  the  definition  given  in  §  132  no  number  can 
have  more  than  one  principal  root  of  any  specified  order. 

133.   Product  of  two  or  more  radicals  of  the  same  order.* 

Just  as  V9  .  V25  =  V^,  i.e.,  V9^^, 

and  ^^^'-y/ 27=^^^216; 

[Each  member  of  the  first  of  these  equations  being  15,  and  of  the  second,  —  6.] 

SO,  too,  if  X  and  y  are  any  numbers  whatever  (cf.  footnote,  p.  229), 
and  n  is  any  positive  integer, 

•\/x  •  -y/y  =  \/xy. 

*  In  §§  133-145  imaginary  numbers  are  excluded,  and  the  proofs  are  further 
limited  to  "  principal  roots." 


132-134]  IRRATIONAL  NUMBERS  .  229 

For,,  since  (-y/x-\/yy  =  {-y/x^y)  •  (-Vx-y/y)  •••  to  n  factors 

=  (^xy  .  (Vyy  [§§  52  and  53 

=  xy; 

i.e.,  since  the  nth  power  of  -Vx  •  ^y  is  xy,  therefore  (§  130) 

Vic  •  -\/y  =  ^xy.  (1) 

Similarly,  it  is  easily  shown  that 

■\/x  •  ^y  •  -y/z  '••  =  ^xyz  •  •  •,  (2) 

which  may  be  formulated  in  words  thus :  the  -product  of  the 
nth  roots  of  two  or  more  numhers*  is  the  nth  root  of  the 
product  of  those  numhers. 

EXERCISES 

Express  each  of  the  following  indicated  products  as  a  single  radical : 

1.  V5-\/7.     ,  4.    \/3a.Vl0te. 

2.  \/3.V7-\/2.  5.    y/¥^  •  Vb^  •  VW^. 

3.  y/2  •  ^6  .  y/l  •  v^.  6.    V^T~y  •  v'^TT^. 


7.  Verify  that  vx  +  y  •  Vx  —  y  =  Vx^  —  y^  when  x  =  5  and  ?/  =  4. 

8.  Is  the  equation  in  Ex.  7  true  for  all  values  of  x  and  y,  or  only  for, 
certain  particular  values,  such  as  a;  =  5  and  y  =  4?    Why?  ( 

9.  Is  Va  •  Vh  equal  to  Va6  ?     Why  ?    If  Va  •  v^i  were  also  equal  to 
Vab,  how  would  Vb  and  Vft  compare  ? 

10.  Is  V&  equal  to  Vb  when  &  ^t  1  ?    Is  then  Va  •  V6  equal  to  Vab  or 
to  Vab  for  all  values  of  a  and  &? 

11.  When  may  the  product  of  two  or  more  radicals  be  expressed  as  a 
single  radical? 

134.   Special  cases  of  §  133.     It  x  =  y,  then  Eq.  (1)  of  §  133, 

viz.,  -Vx  •  -Vy  =  '\/xy, 

becomes  a/cc  •  ^x  =  ^xx, 

i.e.,  ('Vxy  =  -y/x^. 

*  If  n  is  even,  these  numbers  must  be  positive,  since  imaginary  numbers  are 
excluded  from  the  present  discussion. 


230  ELEMENTARY  ALGEBRA                      [Ch.  XIV 

Similarly,  if  x  =  y  =  z=-';  then  Eq.  (2)  of  §  133, 

viz.,  ^/x'  ^y  '  ^z  •"  =  ^xyz  •••, 

becomes  (-v/^)^  =  V^,                                       (1) 

where  jp  is  any  positive  integer,  i.e.,  the  pth  poiver  of  the  nth 
root  of  a  number  is  equal  to  the  nth  root  of  the  pth  power 
of  that  number. 

Again,  if  either  ic  or  ?/  is  itself  the  nt\i  power  of  some  number, 
say  X  =  a",  then  Eq.  (1)  of  §  133, 

viz.,  Vx  •  Vy  =  Vxy, 

becomes  Va"  •  -^y  =  -y/a^'y, 

i.e.,  a^y  =  Va^'y ;  (2) 

hence,  a  coefficient  of  a  radical  may  be  inserted  (as  a  fac- 
tor) under  the  radical  sign  by  first  raising  it  to  a  power 
corresponding  in  degree  to  the  index  of  the  root ;  and  (read- 
ing Eq.  (2)  from  right  to  left)  a  factor  of  the  radicand,  which 
is  an  exact  power  corresponding  in  degree  with  the  indi- 
cated root,  may  be  placed  outside  of  the  radical  sign  (as  a 
coefficient)  by  merely  extracting  the  indicated  root. 

EXERCISES 

1.  What  is  the  value  of  (Vi)^?    Of  v/i^?    How,  then,  does  (\/4)8 
compare  with  \/48  ?     Does  this  agree  with  Eq.  (1)  above  ? 

2.  Is   (v^)6  equal  to  -t/75?     Why? 

3.  What  is  the  value  of  5  v^  ?   Of  v/pTs,  i.e.,  of  v^lOOO?    How,  then, 
does  5\/8  compare  with  VS^  •  8 ?    Does  this  agree  with  Eq.  (2)  above? 

4.  Is3\/5equalto  V32T5?    Why? 

5.  Using  the  method  by  which  Eq.   (2)  above  was  established,  prove 
the  correctness  of  your  answer  in  Ex.  4. 

In  the  following  expressions  insert  the  coefficients  under  the  radical 
signs,  and  explain  your  work  in  each  case : 


134-135] 

IRRATIONAL 

NUMBERS 

2; 

6.   3v/5. 

10.  fVB. 

14. 

W2i. 

7.   2VI0. 

a  2^. 

11.  |Vf|. 

12.  fVSaa: 

15. 

x  +  i^j     3 
x-i  ^^  x  +  i 

9.    5^4. 

13.    ^^/l2 

a^a:. 

16. 

±-y/a^x{x-  \). 

17.  State  in  words  how  a  coefficient  of  a  radical  may  be  inserted  under 
the  radical  sign. 

Write  each  of  the  following  radicals  in  a  form  having  the  radicand 
as  small  as  possible  : 

18.  Vi5. 

Suggestion.     \/45  =  VsTTB  =  V32T5,  —compare  Eq.  (2)  above. 

19.  Vl80.  23.    y/-  192. 

20.  vT62.  24.    V892a8^. 

21.  ^^'320,  25.    y/imd^¥x^^  ^^       ,^—, 

30.    V3  x^  -Qxy  ^-6  y\ 

22.  \/-  54.  26.    ^-486mV.  31.    12v/-8m4+ 24m8n. 
32.   Is  V^  equal  to  a;?/  ?     Why? 


27. 

V12«3(a:  +  2/)5. 

28. 

v/ltja^a:*- 246x6. 

29. 

V18  a  -  9. 

33.  Is  Va;2  +  ?/2  gq^a]  to  a;  +  ?/?     Why? 

34.  Verify  your  answer  to  Ex.  33  when  x  =  3  and  y  =  4. 

35.  Is  the  extraction  of  roots  distributive  over  a  sum  ?     Over  a  prod- 
uct?   Compare  Exs.  32  and  33. 

135.   Quotient  of  two  radicals  of  the  same  order. 

Just  as  .  |=  =  \/|'  [Each  being  I 

SO,  too,  if  X  and  y  are  any  numbers  whatever  (cf .  footnote,  p.  229), 
and  n  is  any  positive  integer. 


«/7.        \  li 


yy 


232  ELEMENTARY  ALGEBRA  [Ch.  XIV 

To  prove  this  it  is  only  necessary  to  remember  that 

('lly^l^.ll.l:!...  ton  factors 
\y/yj       y/y    Vy    -Vy 


-y/g?  ♦  -\/x  »  -y/x  » •  •  to  n  factors 
■\/y  .  ^y  .  -s/y  ...  to  n  factors 


[§  54  (ii) 


(Vyy   y' 

i.e.,  the  nth  power  of  ^^  is  -,  and  therefore,  by  the  definition 

^y     y 

of  a  root  (§  130),   ^^=a/-, —  which  was  to  be  proved. 
Vy     ^y 

The  student  may  state  in  words  what  has  just  been  proved 
(cf.  §  133). 

EXERCISES 

Express  each  of  the  following  quotients  by  means  of  a  single  radical : 

1.  V35--V5.  4.    \/lQ  a^x^  -~  VW^K 

2.  \/216-^Vl2.  5.    v'a;2  -  y^  ^  V^+^. 

3.  \/216-\/l2.  6.    \/16  a^b^  -  32  a^x^  -  \/4a2. 


7.  Verify  that  Va^  -  6^  ~  y/a  -h  =  \/a  -\-  b  when  a  =  5  and  &  =  3. 

8.  Is  the  equation  in  Ex.  7  true  for  all  values  of  a  and  b,  or  only  for 
certain  particular  values  of  these  letters  ?    Why  ? 

9.  Is  y/W^^^Wai  equal  to    ^/^rt^.^     ^j^^.     Compare   also 
Ex.  9,  §  133.  \  Qax 

10.   If  two  radicals  are  of  different  orders,  can  their  quotient  be  ex- 
pressed as  a  radical  of  the  same  order  as  either  one  ? 


11.    iP^'^  :  -i/l   ^   =? 


12    * l^^^y^ ^  5  /__2rt_ 

'    \    35  68     ■   \1b'^xh 


3  a  \    35  68        \7  62xV 


135-136]  IRRATIONAL  NUMBERS  233 

136.  Radicals  whose  indices  are  composite  numbers. 

Just  as  </M  =  -y/V^  =  V^,  [Each  being  2 

so,  too,  if  X  is  any  number  whatever  (cf .  footnote,  p.  229),  and  n 
and  p  are  positive  integers, 


This  principle  may  be  proved  as  follows  (cf.  §§  133  and  135) : 
\^-\/x)    =  V -^/a; .  V-v/a; .  V^x  •••to  np  factors 
=  i  V^ .  V^ .  i/^ ...  toi>  factors i" 

=  I  V^J«  [Since  ( V VS)"  =  V^ 


i.e.,  the  n^th  power  of    \^x  is  ic,  and  therefore    \-\/x=^^- 
In  the  same  way  it  may  be  shown  that  '^Vx  =X^x. 

This  principle  is  useful  in  extracting  roots  whose  indices  are 
composite  numbers  (cf.  §  129,  note). 

EXERCISES 

1.  What  is  meant  by  the  symbol  VN  (cf.  §§  122  and  130)  ?    Point 
out  two  places  in  the  above  proof  where  this  definition  is  employed. 

2.  Using   §  136,   show  that   </T25=V5,  and   y/M=</E;   also  state 
verbally  the  general  principle  which  is  involved  in  these  equations. 

Reduce  each  of  the  following  radicals  to  an  equivalent  radical  of 
lower  order: 

3.  v^.  5.  \/3i3.  7.    \/'d2a^b^°xK  9.  \/l21  aV- 

4.  v^.  6.  Wfx^.         8.  y/a*b^xY,  10.  </^^  -  2  QX -\- x^. 


234  ELEMENTARY  ALGEBRA  [Ch.  XIV 

137.   Changing  the  order  of  a  radical.     It  follows  directly  from 
the  principle  established  in  §  136  that 


wherein  a  is  any  number  whatever  (cf.  footnote,  p.  229),  and  w, 
p,  and  t  are  positive  integers ; 

for,  Vi^='V(^  [§121(ii) 

[§136 

[Since  </N''=N 

that  is, 

hence,  multiplying  both  the  index  of  tim  radieal  and  the 
exponent  of  the  radicand  hy  any  positive  integer,  or  divid- 
ing them  both  hy  any  positive  integral  factor  which  tJiey 
may  contain,  leaves  the  value  of  the  expression  unchanged.* 

EXERCISES 

1.  Is  Vofi  equal  to  Va?    Why?    ^Employ  §  136  to  prove  the  correct- 
ness of  your  answer.     Show  also  that  it  follows  from  §  137. 


2.  Is  V  3  a^x  equal  to  wd  a^x^'i  Why?  Show  that  the  correctness  of 
this  equation  follows  from  §  136 ;  also  from  §  137. 

3.  Reduce  Va^i^  to  an  equivalent  radical  of  the  second  order,  and 
explain.     Also  Vx^y^^. 

4.  Change  v2a;*  to  an  equivalent  radical  of  the  12th  order,  and 
explain.     Also  y/ahf. 

5.  Reduce  V25  mH^  and  VS  a^ft^x^,  respectively,  to  equivalent  radicals 
of  the  fourth  order,  and  explain. 

6.  Reduce  V3ax,  \/2m'^n,  and  Vd^-n^x^^  respectively,  to  equivalent 
radicals  of  the  12th  order ;   of  the  24th  order. 

7.  Write  out  a  carefully  worded  rule  (from  the  principle  of  §  137)  for 
reducing  given  radicals  to  equivalent  radicals  of  higher  or  lower  orders, 
and  state  the  necessary  limitations. 

*  This  property  at  once  suggests  that  the  exponent  of  the  radicand  and  the  index 
of  the  root  bear  to  each  other  a  relation  similar  to  that  of  the  numerator  and 
denominator  of  a  fraction ;  this  relation  will  be  more  fully  considered  in  §  153. 


137-138]  IRRATIONAL  NUMBERS  235 

138.  Reduction  of  radicals  to  the  same  order.  Comparison  of 
radicals.  From  §  137  it  follows  that  any  two  or  more  radicals 
(real  numbers)  may  be  reduced  to  radicals  which  are  of  the  same 
order,  and  which  are,  respectively,  equivalent  to  the  given  radicals. 

E.g.,  VE  and  Vt  are  respectively  equivalent  to  ^'\/5^  and  *'\/7^,  i.e.,  to  v''625 
and  v/3i3. 

The  student  may,  by  this  method,  reduce  VA  and  a/B  to 
equivalent  radicals  of  the  same  order ;  he  may  then  formulate  the 
procedure  into  a  rule. 

Reducing  any  two  given  radicals  (real  numbers)  to  the  same 
order  furnishes  a  means  for  comparing  the  values  of  these  radicals  ; 
thus,  in  the  above  illustration,  S/5  >V7  because  their  respective 
equivalents,  viz.,  ^625  and  \/343,  stand  in  this  relation. 

EXERCISES 

Reduce  the  following  to  equivalent  radicals  of  the  same  order,  and 
thus  compare  their  values : 

1.  V5  and  y/VL.  3.   \/lO,   V2,  and  v^S. 

2.  \/7  and  V3.  4.  Vl,  v^,  and  ^5. 

Reduce  the  following  to  equivalent  radicals  of  the  same  order : 
5.  V3a6,  y/2^\  and  y/^a%H^  7.   y/x,  V^j,  and  V^. 


6.   \/2x%  y/ax,  and  v2m%.  8.  Va  +  6,  Va^  +  ft^^  and  Va  —  b. 

9.  Can  the  radicals  in  Ex.  5  be  reduced  to  equivalent  radicals  of 
the  6th  order?  Of  the  12th  order?  Of  the  9th  order?  Give  the  reasons 
for  your  answer  in  each  case. 

10.  What  is  the  lowest  common  order  to  which  the  radicals  in  Ex.  6 
can  be  reduced  ?    Those  in  Ex.  7  ?     Those  in  Ex.  8  ? 

11.  Compare  the  rule,  asked  for  in  §  138,  with  the  procedure  in  solving 
Exs.  1  to  8,  and  see  whether  it  meets  all  the  requirements. 

12.  Which  is  greater,  3V5  or  2v/IT?     Compare  §§  134  and  138. 

13.  Whichisgreater,  2v/9  or3V3\/2?    Why? 

14.  How  may  the  values  of  any  two  numerical  radicals  (real  numbers) 
whatever  be  compared  ? 


vn 


T 


236  ELEMENTARY  ALGEBRA  [Ch.  XIV 

139.  Reduction  of  radicals  to  their  simplest  forms.  A  radical  is 
said  to  be  in  its  simplest  form  when  the  radicand  is  integral,  when 
the  index  of  the  root  is  as  small  as  possible,  and  when  no  factor 
of  the  radicand  is  a  perfect  power  corresponding  in  degree  with 
the  indicated  root. 

The  following  examples  may  serve  to  illustrate  the  application 
of  the  foregoing  principles  to  the  reduction  of  any  given  radical 
to  its  simplest  form. 

Ex.  1.     Reduce  v^f  to  its  simplest  form. 

Solution.  ^  =  y|^  =  ^F^  =  j  ^.  [§  134 


Ex.  2.     Reduce  Vi  a^x^y^  to  its  simplest  form. 

Solution.  y/Ia^xY  =  v^(2a^VF  =  ^2  axY-  [§  137 


Ex.  3.     Reduce  v8  a^x^y^  to  its  simplest  form. 
Solution.  VsS^  ^  v/4 a^xY  •  ^2ax  [§  133 

=  2  ax^y  V2  ax. 

EXERCISES 

4.  Is  VS  ax  in  its  simplest  form  ?     Why  ? 

5.  Is  V12ax  in  its  simplest  form?     Why? 

6.  Is  5  ViaW  in  its  simplest  form ?    Why ? 

7.  Is  12  Vf  ax^  in  its  simplest  form  ?    Reduce  it  to  its  simplest  form. 

8.  What  is  meant  by  saying  that  a  radical  is  in  its  simplest  form? 

Reduce  each  of  the  following  radicals  to  its  simplest  form : 

9.  Vl2.                          19.    \/|.                                 26.   3  \/25  a%^x^ 
10.    Vl62.  20    A  r^  «/ 

11.  v/16.  ,  "^  y 

21.    v^. 


12.    ^250.  I—.  28.    ^a'+%2.y.-4. 


13.  \/81.  •   A/  9  ^^2*  29.    \/a2na;»+5. 

14.  ^189.  ._    ^/TTift^  3 — 

15.  VI2-8.  "'-^W'  30.    V-40.^-B,H. 

16.  </32.  -  24.  V^±J.  31.   4  ^^ 

17.  ^640. 


a^"a:". 


18.   Vj. 


25.   3aV^^=^.  32.   -^I^Tli. 


139-140]  IBBATIONAL  NUMBERS  237 

140.  Addition  and  subtraction  of  radicals.  Similar  radicals  (cf. 
§  131)  may  evidently  be  added  and  subtracted  just  as  rational 
numbers  are  added  and  subtracted,  i.e.,  by  regarding  the  common 
radical  factor  as  the  unit  of  addition. 

E.g.,  just  as  3  + 10  —  4  =  9,  in  which  1  is  the  unit,  and  3  a  + 10  a  —  4  a  =  9  a,  in 
which  a  may  be  regarded  as  the  unit  of  addition,  so  3\/2  + 10\/2  —  4V2  =  9 V2, 
in  which  V'2  may  be  regarded  as  the  unit  of  addition. 

If  the  radicals  are  in  their  simplest  forms  and  are  dissimilar, 
then  their  sum  or  difference  can  only  be  indicated,  and  this  is 
done  by  connecting  them  with  the  proper  signs. 

E.g.,  the  sum  of  7n/15,  ^aV^xy^,  and  3\/2  is  indicated  thus: 
7  \/l5  +  3  a  v^2^  +  3  \/2 . 

If  the  radicals  which  are  to  be  added  are  not  in  their  simplest 
forms,  they  should  first  be  reduced ;  the  following  examples  may 
serve  to  illustrate  the  procedure : 

Ex.  1.     Find  the  sum  of  V75  and  3  \/l2. 

Solution.     V75  +  3  >/l2  =  V25  •  3  +  3\/4T3'=  5\/3  +  6V3  =  ll\/3. 

Ex.  2.     Find  the  sum  of  5  vTS,   -  VOS,  and  ^/\. 


Solution.     5  Vl8  -  Vo.5  +  Vi-  =  5  Vo  •  2  -  V^  .  2  +  V  Jg  •  2  * 

=  15  V2  -  ^  V2  +  i  V2  =  14f  \^. 


Ex.  3.     Find  the   sum   of   V9  ar  -  18,   6  V4  x  +  8,    \/36  x  -  72,   and 
-  \/25  X  +  .50. 


Solution.     Vq  x  -  18  +  6  \/4  x  +  8  +  \/36  a:  -  72  -  V25  x  +  50 


3  Vx^^  +  12Vx  +  2  +  6\/x-2-5\/a:  +  2 


=  9  Vx  -  2  +  7  Vx  +  2. 

EXERCISES 

Find  the  sum  of: 

4.  VSO,  Vl8,  and  \/98.  6.    \/28,  V63,  and  VTOO. 

5.  \/l2,  V75,  and  V27.  7.    \/250,   v^,  and  ^/U. 

8.    v/500,   v^l08,  and  V^^^. 

*  Since  0.5  =  ^  =  f ,  and  ^  =  i^. 


238  ELEMENTARY  ALGEBRA  [Ch.  XIV 

9.    What  is  the  sum  of  a,  2  b,  and  c  ?    Of  3  x,  4  y,  a,  2  a:,  and  —  5  y  ? 

10.  What  is  the  sum  of  3  V2  and  Sv'T?  Of  3  V2,  5  v^,  -  2  V7, 
and  V2? 

11.  Write  out  a  carefully  worded  rule  for  the  addition  and  subtraction 
of  radicals ;  provide  both  for  those  cases  in  which  the  given  radicals  are 
similar  and  for  those  in  which  they  are  dissimilar. 

Simplify  the  following  expressions  as  far  as  possible,  and  explain  your 
work  in  each  case : 

12.  v/135  +  v^625  -  v^320.  18.    vl28¥  +  \/375^  -  v^547. 

13.  ^+V2-8  +  Vl7-5  +  ^.  ^^    J-l_,JJ_J1, 

14.  v^375  -  Vii  -  v/192  +  V99.  '   ^^^      ^^^     ^2' 

15.  V1+V75-V/12  +  1V3.  ^^   ^1^2^,       iyq—^       ra 

16.  Vl47-x4  +  iV3  +  ^^9.  '   ^^V      ^    bf        ^hf' 

17.  6  \^  +  4  v^JI  -  8  \^.  21.    V(a  +  b)c  -  V(^^^h)^. 


22.  v/192  a;4  -  2  V3  a;4  -  v/5  a:  +  V40  x*. 

23.  </7^  +  V^b^  -  y/8^^b^. 


24.    V3  a:3  +  30  x^  +  75  a;  -  V'Sx^  -  Q  x^  + 'd  x. 


25.  \/5  a6  +  30  a"  +  45  a^  -  V5  a^  _  40  a*  +  80  a^ 

26.  V50  +  v^  -  4  V|  +  v^:r24  +  v^  -  ^^64. 

27.  Vf  +  6  Vf  -  Wl8  +  v^36  -  v^  +  Vl2b  -  V^ 


28.    Va3  -  a^x  -  Vax^  -  x^  -  V(a  +  x)  (a^  -  x^). 

141.  Multiplication  of  monomial  radicals.  In  §  133  it  is  shown 
how  to  get  the  product  of  two  or  more  radicals  which  are  of  the 
same  order,  and  in  §  138  it  is  shown  how  to  reduce  any  given 
radicals  to  the  same  order ;  therefore  the  product  of  any  two  or 
more  monomial  radicals  (real  numbers)  may  now  be  found. 

Ex.  1.    Multiply  \/5  by  V2. 

Solution.  4^5  •  \/2  =  v/p  •  \/2'^  [§  188 


v/52 .  28  =  V200.  [§  133 


140-142]  IBRATIONAL  NUMBERS  239 

Note.  The  student  should  observe  that,  although  a  root  remains  to  be  ex- 
tracted in  this  resjilt,  viz.,  V'iOO,  the  result  is  simpler  in  form  than  the  indicated 
product,  viz.,  Vs  •  \/2,  and  also  that  the  arithmetical  work  of  finding  the 
approximate  numerical  value  is  much  easier  in  the  final  than  in  the  original 
form. 

Ex.  2.    Find  the  product  of  5V2  by  8v^7. 

Solution.         5\/2  •  8\/7  =  5  •  8  .  V2  •  v^Y  [§52 

=  40\/23.  v^2^40\/392. 


EXERCISES 

3.  Multiply  V3  by  VG,  and  simplify  the  result. 

4.  Multiply  \/3  by  \^. 

5.  How  may  the  product  of  two  or  more  radicals  which  are  of  the 
same  order  be  found  (cf.  §  133)  ? 

6.  How  may  the  product  of  two  or  more  radicals  which  are  of  differ- 
ent orders  be  found  ? 

Find  the  following  products,  and  simplify  the  results  : 

7.  V3  by  Vl5.  15.    VxY  '  ^^12^  •  ^^^Wxf. 

8.  2 V5  by  3Vl0.  16.    V2^  •  VWc  •  y/r^\ 

9.  5V2  by  4v^.  17.    VFi^.  V^F^^ .  V^=Y. 

10.  \/3  by  3\/3.  18.    V^^  by  \/8a8. 

11.  2\^  by  7v^l0.  19.    ^a'  by  V^\ 

12.  2^2  by  'v/512.  20.   3v/2  by  ^V%  i.e.,  (3v^2)2. 

13.  v/|  by  2\/|.     .  21.    (2v^5"^)8. 

14.  y/2'^'-^.  22.    (v^l2a-2xV)». 

142.   Multiplication    of    polynomials    containing    radicals.      The 

product  of  two  polynomials  containing  radicals  is  obtained  by 
multiplying  each  term  of  the  multiplicand  by  each  term  of  the 
multiplier  and  adding  the  partial  products,  just  as  in  the  case  of 
rational  polynomials. 


240  ELEMENTARY  ALGEBRA  [Ch.  XIV 

Ex.  1.   Multiply  5V2  -  2V3  by  3  V2  +  4V3. 
Solution.  5V2  — 2a/3 

3\/2  +  4V3 
30-6V6 

+  20\/6-24 
30  +  14V6-24  =  6  +  14\/6. 

Esc.  2.   Expand  (2  \/3  —  V2)2  by  the  binomial  theorem. 
Solution.     (2V3  -  ^2)2  =  (2\/3)2- 2(2\/3)  v^2 +  (v/2)2  [§57 

=  12-4v/l08  +  v^.  [§140 

EXERCISES 

Perform  the  following  multiplications,  and  simplify  the  results : 

3.  V5-5  by  Vo  +  1.  7.   2V3  +  v^2  by  2V3  -  y/l. 

4.  2\/2+V3  by  \/2  +  4V3.  8.    a'^-abV2  +  b^  by  a2+a6V2+>. 

5.  v^2  +  3  v/2  by  Vi.  9.   x  ~  -Vxyz  +  yz  by  V^  +  Vyz. 

6.  5  +  v/4  -  2  v/5  by  \/5  +  \/6.         10.    -Va  +  Vxby   -Va-Vl. 

Expand  the  following  expressions,  and  simplify  the  results : 

11.  (V2-3v/3)2.  14.  (v;^r^  +  v;;H^)2. 

12.  (\/2^-V3^)2.  15.    (v/'J^- ^3^2)3. 

13.  (a  +  V6  -  \/c)2.  16.    (v/a  +  2\/3)5. 

18.   (V2a-V6  +  V2a  +  V6)l 

143.  Division  of  monomial  radicals.  By  means  of  §§  135  and 
138  the  quotient  of  any  two  given  monomial  radicals  (real 
numbers)  may  be  expressed  as  a  single  radical  (cf.  §  141). 

Ex.  1.    Divide   v^4  ax^f  by   VTcih:. 

Solution.  V±^^  _'^^W^^ 


V2  a^x  V8  a^xs 


[§138 


=  :MJ^^  [§135 

=  a/?^-^'  =  1  'V2  a^x^.  [§  139,  Ex.  1 


142-144]  IRRATIONAL  NUMBERS  241 

EXERCISES 

2.  What  is  the  quotient  of  V50  divided  by  V8  ? 

3.  What  is  the  quotient  of  4V5  divided  by  VIO? 

4.  What  is  the  quotient  of  1  Vbi  divided  by  2v^686V 

5.  How  is  the  quotient  of  two  monomial  radicals  obtained  if  these 
radicals  are  of  the  same  order  ? 

Express  each  of  the  following  indicated  quotients  in  its  simplest  form  : 

6.  2v^-\/8.  »  11.    Voi-^V^. 

7.  2^6 -^v^.  12.    V2^-^^o"^2^. 

8.  Vl8  -  \/500.  13.   2  v/9a2p  ^  3 V3^. 


14.   a\/^x^y'-^2b</'2xy. 


10.    \/|-3v/f.  15.   3aV2^^«^i-26v/3a:'*-5. 

16.  How  is  the  quotient  of  two  monomial  radicals  obtained  if  these 
radicals  are  of  different  orders  ? 

17.  Apply  the  answer  of  Ex.  16  to  show  that 

y/x'^  -  y^  -f-  vT+l/  =  -^  y/ ix  -  y)\x ->r  yY  =  -^  ^(x2-?/2)2(x  +  3/)8. 
x+y  x+y 

Verify  this  equation  when  x  =  64  and  ?/  =  0.  Is  this  equation  true  for 
all  values  of  x  and  y,  or  merely  for  certain  particular  values  of  these 
letters  ?     What  other  name  is  given  to  such  equations  (cf.  §  23)  ? 

144.  Division  of  polynomials  containing  radicals.  If  the  divisor 
is  a  monomial,  then,  manifestly,  the  quotient  may  be  obtained  by 
dividing  each  term  of  the  dividend  by  the  divisor  —  just  as  in  the 
case  of  rational  expressions. 

E.g.,  3i^+ivl^lW„3+4Ji-2^|  K138 

=  3  +  2V6-2v^2.  [§139 

Instead  of  dividing  directly  by  a  radical,  it  is  usually  advan- 
tageous first  to  multiply  both  dividend  and  divisor  by  an  expres- 
sion which  will  make  the  new  divisor  rational  —  indeed,  it  is 
frequently  necessary  to  do  so. 


242  ELEMENTARY  ALGEBRA  [Ch.  XIV 

E.g.,  since        (3\/2-\/i3)  •  (3 a/2  +  VlS)  =  (3>/2)2-  (Vi3)2  =  5, 

therefore  5  -^  (3  V2  -  Vi3)  =  3  V2  +  Vi3 , 

but  oue  could  not  easily  obtain  this  quotient  by  dividing  directly.    It  may  be 
obtained  thus: 

5(3V2+\/l3)  pMultiplying  numerator  and 


3V2— Vl3      (3V2  — Vl3)(3V2  +  Vl3)  L  denominator  by  3  V2+\/i3 

^16vl+5Vl3^3^2+Vl3. 

This  method  of  dividing  (usually  called  division  by  means  of 
rationalizing  the  divisor)  will  often  be  found  very  advantageous 
even  when  it  is  not  strictly  necessary. 

jEq  3V2  +  4v/3^(3V2+4\/3)  .  V2^6  +  4V6^  3  ,  g^/g 

V2  (\^)2  2 

The  factor  by  which  a  given  radical  is  multiplied  to  obtain  a 
rational  product  is  called  its  rationalizing  factor. 

E.g.,  of  v^4  and  ^2  each  is  a  rationalizing  factor  of  the  other  (why?) ;  so  also 
are  Vop  and  \/o"-p  (why?),  and  aVx  +  hy/y  and  aVz  —  hy/.y  (why?).* 

Of  two  such  binomial  quadratic  surds  as  a-\/x  +  &Vy  and 
a^x  —  h^y,  which  differ  from  each  other  only  in  the  quality 
sign  of  one  of  their  terms,  each  is  called  the  conjugate  of  the  other. 


EXERCISES 

1.  Divide  Vl5  -  V3  by  V3. 

2.  Divide  V6  +  2  V3  by  \/2. 

3.  Divide  v^  -  4  V5  +  2  v^G  by  V3. 

4.  Perform  the  divisions  in  Exs.  1-3  by  first  rationalizing  the  divisors, 
and  show  whether  or  not  there  is  any  advantage  here  in  rationalizing. 

5.  Show  that  2  VB  -  Vs  is  a  rationalizing  factor  of  2  VS  +  \/5. 

6.  Is  \/5  -  2  \/3  a  rationalizing  factor  of  2\/3  +  VB?     Why?     Are 
these  surds  conjugate  to  each  other? 


*  The  question  of  finding  rationalizing  factors  for  given  expressions  is  further 
considered  in  §  161. 


144-145]  IRRATIONAL   NUMBERS  243 

Find  the  simplest  rationalizing  factor  of  each  of  the  following  surds  : 


11.5^ 


7.    V2a.  „     ^12ahn  15.   3a -2" 


ox. 


8.  V4^2.  L    ^    _  ^^-  oa:-\/2^. 

3, ^2.    V2-V7.  17^  v^+2V3-6. 

9.  V4ax-^.  ^3^   2V8+V6.  /^^^ 

18.  ^ii£  +  |v^8. 


10.    Va  +  6.  14.   4  +  5  V3.  ^    a 

19.  Divide  31  by  7  +  3  V2. 

20.  Divide  2  V6  by  V5  -  VB. 

21.  Divide  5\/l2  —  2  V6  +  4  by  \/4.  What  is  the  smallest  multiplier 
that  will  rationalize  v^l  ? 

22.  Divide  3  ■\/2  -  4  V5  by  2  V3  +  V7. 

23.  Divide  4V3  +  5  V2  by  3  V2  -  2  VB. 

24.  If  the  result  of  Ex.  21  were  wanted  correct  to  4  decimal  places, 
say,  show  in  detail  that  it  is  far  simpler  first  to  rationalize  the  divisor 
than  to  extract  roots  and  divide  by  the  ordinary  arithmetical  method. 

25.  What  is  the  product  of  (2  +  VB)  -  V5  by  (2  +  V3)  +  V5  ?  Of  this 
result  by  2  —  4  \/3  ?   What  then  is  a  rationalizing  factor  of  2  +  V3  —  V5  ? 

Of  2  +  V3  +  V5  ? 

26.  Divide  2  -  V3  by  1  +  V3  -  \/2. 

Reduce  the  following  to  equivalent  fractions  having  rational  denomi- 
nators : 


a  +  Vq^  +  ar  ga     ^^  +  ^  -y/x  -  y.  39     E!_ 


27.   "  ^  "  "•   ^  •".  28.     "-^  ^  !f l^ ^.  29. 


a  —  V  a^  +  X  Vx  +  2/  +  Vx  —  y  Va^  +  z^  —  z 

30.   Simplify  -^  +  -^_.  31.   Simplify  (V2  +  3)(v/5-2), 

^_1      ^  +  1  (3-V2)(2  +  V5) 

32.    Find  the  value  of :;  -| ^^^  correct  to  3  decimal  places. 

2  -  V3      \/2  +  1 

145.  An  important  property  of  quadratic  surds.  Neither  the 
sum  nor  the  difference  of  two  dissimilar  quadratic  surds  (§  131) 
can  be  a  rational  number ;  for,  if  possible,  let 

■y/x-\-Vy  =  r,  (1) 

Vx  and  Vy  being  dissimilar  surds,  and  r  rational,  and  not  zero. 


244  ELEMENTARY  ALGEBRA  [Ch.  XIV 

From  Eq.  (1)  Vy==r  —  Vx,  (2) 

whence,  squaring,  y  =  r^  —2  rVx  +  x,  (3) 

and,  solving  for  Vx,  Vx  =  — -^-- ^, 

I.e.,  if  Eq.  (1)  were  true,  then  the  surd  Vx  would  equal  the  rational 

number  —  ,^  ~  ^ ,  which  is  impossible;  hence  Eq.  (1)  can  not  be 
true. 

Similarly,  Vx  —  Vy  ^  r. 

From  what  has  just  been  shown  it  at  once  follows  that 
if  x-\-  Vy  =  a  +  Vb,  where  x  and  a  are  rational,  and  Vy 
and  Vb  are  quadratic  surds,  then  x  =  a  and  y  =  b. 

For,  if  x-{-  Vy  =  a  +  Vb, 

then  Vy  —  Vb  =  a  —  x; 

which,  by  the  above  proof,  can  be  true  only  if  each  member  is 
zero,  i.e.,  if  a  =  a;  and  Vy  =Vb.  In  other  words,  the  equation 
x-\-Vy  =  a-\-Vb  is  equivalent  to  the  two  equations  x  =  a  and 
y  =  b. 

II.     IMAGINARY  NUMBERS 

146.  Imaginary  numbers.  In  solving  the  equations  of  the  next 
chapter,  indicated  square  roots  of  negative  numb'ers  frequently 
appear;  such  numbers  have  already  been  defined  (§  130)  as 
imaginary  numbers ;  if  they  present  themselves  in  the  form  V—b, 
where  6  is  a  positive  number,  they  are  called  pure  imaginary  num- 
bers, while  if  they  present  themselves  in  the  mixed  binomial  form 
a -{-V—b,  where  a  and  b  are  real  and  b  is  positive,  they  are  usually 
called  complex  numbers.* 

*  A  broader  definition  of  imaginary  numbers  is  given  in  appendix  B,  where 
it  is  shown  that  every  such  number  can  be  expressed  in  the  form  a  +  bV—l, 
and  where  it  is  proved  that  these  numbers  obey  the  laws  already  established  for 
real  numbers  (commutative,  associative,  etc.).  Logically  this  proof  should  now 
be  read,  but  it  may  be  deferred  until  later  if  the  reader  will  carefully  bear  in 
mind  that  the  following  discussion  assumes  that  imaginary  numbers  are  subject 
to  those  laws,  and  is  therefore  to  be  regarded  as  tentative  until  this  fact  is  proved. 
The  very  elementary  discussion  which  is  given  in  the  next  few  pages  will  suflSce 
for  present  needs. 


145-147]  IMAGINARY  NUMBERS  245 

E.g.,  \/—5,  2\/— G,  and  V— ^  are  pure  imaginary  numbers,  while  2  — V— 3 
and  7  +  2\/—  5  are  complex  numbers. 

Operations  with  imaginary  numbers  are  greatly  simplified  by 
observing  that,  by  the  definition  of  v  a,  §  130, 

(V^y^-b,  (1) 

and  also  (cf.  method  of  §  133,  and  apply  §§52  and  53)  that 

V^b  =  -Vb'V^.  (2) 

The  symbol  V—  1  is  called  the  imaginary  unit,  and  is  often 
represented  by  the  letter  i. 

147.   Positive  integral  powers  of  V—  1.     As  a  special  case  of 

consequently,  (V—  1)^  i.e.,  ( V—  1)^  •  V—  1  =  —  V—  1. 
Similarly, 

( v^^)*  =  (V^^y .  v^^ = -  v^T  •  v=^ = -  ( v^'  =  1, 

(V^y  =  (V^riy .  ( v^^  =  - 1, 
( V^^)^  =  ( V^i)^  •  ( V^«  =  -  V^^, 

and  so  on  for  the  higher  powers,  i.e.,  the  positive  integral  powers 
of  V—  1  have  only  these  four  values :  V—  1,  —  1,  —V—  1,  and 
1 ;  see  also  Exs.  5,  6,  and  7  below. 

EXERCISES 

1.  Define  an  imaginary  number;  compare  §  130. 

2.  Which  of  the  following  are  imaginary  numbers :    V—  3,    v^—  2, 

^36,  V5,   ^/^^^,  3v^^,  4a-J-^  and  2  +  i  V^s? 

3.  Is  V—  a;  imaginary  when  x  represents  a  positive  number?    When 
X  represents  a  negative  number  ? 

4.  Show   that    if    i  =  V^^,    then    i^  =  -l,    i^  =  -  i,    i*  =  1,    i^  =  i, 
t^  =  —  1,  f  =  —  i,  i^  =  1,  and  i^  =  i. 


246  ELEMENTARY  ALGEBRA  [Ch.  XIV 

5.  Since  any  even  number  may  be  written  in  the  form  2  n,  where  n  is 
an  integer,  and  since  a^"  =  (a^)",  show  that  every  even  power  of  i  is  real. 

6.  As  in  Ex.  5,  show  that  every  odd  power  of  i  is  either  i  or  —  i. 

7.  Since  x"+'^  =  x"  •  x*,  and  since  any  positive  integer  whatever  can  be 
represented  by  one  of  the  following  expressions,  viz.,  4  n  +  1,  4  n  +  2, 
4  n  4  3,  or  4  n,  show  that  the  positive  integer  powers  of  i  can  have  no 
other  values  than  i,  —  1,  —  i,  and  +  1,  and  that  these  values  always  recur 
in  this  order. 

8.  Distinguish  between  pure  and  complex  imaginary  numbers,  and 
give  three  examples  of  each. 

148.  Addition  and  subtraction  of  imaginary  numbers.  By  first 
writing  the  imaginary  numbers  in  the  form  a  +  ftV— 1,  these 
numbers  may  be  added  and  subtracted  exactly  as  though  they 
were  real ;  this  is  illustrated  below. 

Ex.  1.     Find  the  sum  of  V—  4,  4 V—  9,  and  V—  25. 
Solution 
x/i:4  +  4v'^+  V:r25  =  2\^^+4  •  3  V^  +  SV"^     [§  146,  Eq.  (2) 
=  (2  +  12  +  5)  V^  [Footnote,  p.  83 

=  19^^. 

Ex.  2.     Find  the  sum  of  3  +  V^T6,  V^^,  and  5  -  V^T^. 
Solution.    3  +  V-  16  +  V^T  +  5  -  \/^9 

=  3  +  5  +  V^Te  -f  v"^^  -  v^ITg  [§  50 

=  (3  +  5)  +  ( V^iTo  +  v/i:i  -  v:r9)  [§  51 

=  8  +  3V31.  [Ex.  1 


Ex.  3.     Simplify  the  expression  a:V— 4+V—  a:^  —  2x— 1— V—  32. 
Solution.    Since        xV^  =  2  xy/^^, 

V-a:2-2a;-l  =  V -  (x  +  1)2  =  (x  +  1)  V^^, 
and  _VZr32  =  -V32.  \/^l  =  _4\/2.  V^ 

therefore  the  given  expression  becomes 

{2  a:  +  (a;  +  1)  -  4V2}  .  V^T,  i.e.,   (3  a:  +  1  -  4\/2)  •  \^^. 
Similarly  in  general. 


147-149] 


IMAGINARY  NUMBERS 


247 


EXERCISES 

Simplify  each  of  the  following  expressions : 
4.   3+\/:r36-(l +2\/^25)+3\/- 


16. 


5.  V-49  +  5\/^=lt-(6  +  2\/^^). 

6.  V^  -  3V~-^  +  6  V^^TS  -  2  V^^27  +  8  +  V^^^T2. 
7. 


.  |_  (9  V-l  +  5-3  V-24)+3V 
8.    V-  16  ah^  +  Virrs  +  9  V5 


18. 


30  -V-  9a2x2  +  \/-a2x2. 


149.  Multiplication  of  imaginary  numbers.  Multiplication  of 
imaginary  numbers  is  also  performed  by  first  writing  these 
numbers  in  the  form  a  +  ftV— 1;  this  is  illustrated  below. 

Ex.  1.     Multiply  V^^  by  V^. 

Solution.      V^  -  y/~^  =  V2  •  V^  •  V5  .  V^l        [  146,  Eq.  (2) 

=  ( V2  .  V5)  (  V^l .  V^^l)    [§§  52  and  53 

'  .  =VT0.(- l)  =  -\/IO. 

Similarly  in  general :     V—  a  •  V—  6  =  —Vab. 

Note.  The  student  should  carefully  observe  that  (§  133)  the  law  for  the  prod- 
uct of  two  radicals,  i.e.,  principal  roots,  does  not  apply  to  the  product  of  two 
imaginai-y  numbers;  according  to  that  law  the  product  of  V—a  •  V—b  would 
be  V{—a)  •  (~6),  i.e.,  y/ah,  and  not  ~Vab.  Errors  of  this  kind  are  easily 
avoided  by  writing  an  imaginary  number  in  the  form  a  +  6  V—  1  before  operating 
with  it. 

Ex.  2.  Multiply  3  +  V^5  by  2  -  V~^. 

Solution.  Writing  these  imaginary  numbers  in  terms  of  the  imagi- 
nary unit,  the  work  may  be  arranged  thus : 

3  -h  V5 .  v^n. 

2  -  V3  .  a/^T 


6  +  2V5.  v^nr 

-  3\/.3  •  v^i 


Vi5(v/irT)5 


6  +  (2\/5-  3V3)  .  V-  1  +V15. 
Similarly  in  general : 
(a  +  V^^)  •  (c  +  V^)  =  ac-  Vbd  H-  (a  Vd  +  cV6)  V^. 


248  ELEMENTARY  ALGEBRA  [Ch.  XIV 

EXERCISES 
Find  the  product  of : 

3.  3 Vr6  by  5V_  12.        6.    Vr^  +  y/ZT^  by  VI~6  -  VTl. 

4.  sVITg  by  2VI^.  7.   3  +  2Vr^  by  5  -  4VZI;. 

5.  2Vi:i  by  V_  4  a%3.     s.    Vrso  _  2Viri2  by  V^s  -  5V^. 

9.  Show  that  the  sum  and  also  the  product  of  a  +  hi  and  a  —  6i 
(wherein  a  and  6  are  real)  is  real.*  Show  that  this  is  also  true  for 
Vri  _  3  and  -  Viri  _  3. 

10.  Prove  that  both  the  sum  and  also  the  product  of  any  two  conju- 
gate complex  numbers  is  real. 

11.  Multiply  Vir^  +  VITft  +  V3^  by  V-  a  -  V_  ft  +  VT^. 

12.  (1  +  Vr5)2  ^9      13.    (2  -  3  iy  =  ?      14.    (2  a  -  3  a;Viri)2  =  ? 

15.  Find    the    product    of    aV_  6  -f  &V—  a,    aV_  a  -f  ftV—  6,    and 

16.  Show  that  —  J  +  h^  —  3  and  —  J  —  i^^—  3  are  conjugates  of  each 
other,  and  also  that  the  square  of  either  is  equal  to  the  other. 

17.  Write  a  rule  for  multiplying  one  pure  imaginary  number  by 
another,  and  compare  it  with  the  rule  for  getting  the  product  of  two 
monomial  surds  of  the  same  order.     Wherein  do  the  two  rules  differ? 

18.  Reduce  — — — ,ZL  •  +  "^-^ — ^  to  its  simplest  form. 

150.  Division  of  imaginary  numbers.  The  simpler  cases  of 
division  of  imaginary  numbers  are  illustrated  by  the  following 
examples : 

Ex.  1.   Divide  V^e  by  V^^. 

S0.„™..        Q  =  |l^  =  ^  =  V|  =  >^.  [§§146,135 

Similarly  in  general : 

V— a        la      Va  /     a        -,   V—  a        I     a 

*  Of  two  complex  numbers  which  differ  only  in  the  sign  of  the  imaginary  term 
each  is  called  the  conjugate  of  the  other  (of.  §  144). 


149-150]  IMAGINARY  NUMBERS  249 

Ex.  2.   Divide  12  +V_  25  by  3  -VZl. 

Solution.     Such  divisions  are  easily  performed  by  rationalizing  the 
divisor  (cf.  §  144),  thus: 

12  +  V-  25  ^  12  +  5Vin^  ^  (12  +  5V^rT)(3+2\/^^) 
3-V^^        3-2V^:i       (3_2\/^I)(3+2V^n) 

^36  +  39  V^^  +  10(V^n)2 
9_4(V-ri)2 

^  26_-|-_39>/^ 

9+4 

=  2+  3v/^T=2  +  \/39. 

Similarly  in  general:  a  +  &  V£l  ^  (a  +  6 V;=l)(c  -  dV^) 
c  +  dV  -  1      (c  +  d V-  l)(c  -  dV-  1) 

_  CTC  +  ^c?  +  (&c  —  ad)  V—  1 
~  c^  +  d^ 

EXERCISES 

3.   Verify  the  correctness  of  the  result  in  Ex.  2  above  by  multiplying 
the  quotient  by  the  divisor. 


4.   Divide  V- 6  +  2V- 

-8by  V^ 

~2, 

5.   Divide  4  by  1  +  i. 

6.  Divide  2  by  i*  +  i\ 

Simplify  the  following : 

7    2-V^S 

g    V2^  -  3  ai 

3  +  V-2 

V2lc  +  2  6t 

8.  5  +  V-4. 

10.  ^«-^'^^ 

5  -  2  i  iV6  +  Va 

11.  Write  a  rule  for  dividing  one  pure  imaginary  number  by  another, 
and  compare  it  with  the  rule  for  finding  the  quotient  of  two  monomial 
surds  of  the  same  order. 

12.  Divide  3  -  V^+ 2  i  by  2  +  V^  ~  V^  (cf .  §  144,  Ex.  25). 


250  ELEMENTARY  ALGEBRA  [Ch.  XIV 

151.  Important  property  of  imaginary  numbers.  Neither  the 
Slim  nor  the  difference  of  two  different  pure  imaginary  numbers 
can  be  a  real  number  (cf.  also  §  145) ;  for,  if  possible,  let 

V^=^- V^=^  =  r;*  (1) 

then,  transposing,         V—  a  =  r  +  V—  &, 

and  squaring,  —  a  =  i'^  +  2  r^—b  —  6, 

whence  V—  6  =    ~^^  ~ — ; 

z  r 

i.e.,  if  Eq.  (1)  were  true,  then  the  imaginary  number  V—  5  would 

equal  the  real  number  -^^ — ^^ — ,  which  is  impossible,  and  hence 
Eq.  (1)  can  not  be  true. 

Similarly  it  may  be  shown  that  V—  a  +  V— 6  ^  r. 

,     From  what  has  just  been  shown  it  follows  that  if 

a;  +  V— 2/ =  a  +  V— &, 

wherein  a  and  x  are  real  and  V—  ?/  and  V—  6  pure  imaginary 
numbers,  then 

x  =  a  and  y  =  h.         , 

For,  if  x  +  ^—y  =  a-\- V— 6, 

then,  transposing,      ^—y  —  V—b  =  a  —  Xj 

which,  by  the  above  proof,  can  be  true  only  if  each  member  is 

zero,  i.e.,  if  y  =  b  and  x  =  a, 

which  was  to  be  proved. 

In  other  words,  the  equation  x  +V—  y  —  a  +V—  b  is  equiva- 
lent to  the  two  equations  x  =  a  and  y  =  b. 


*  The  expressions  V—  a  and  V—  6  represent  different  pure  imaginary  num- 
bers, and  r  is  real,  and  not  zero.  .    . 


151-152]  IMAGINARY  NUMBEBS  251 

152.  Complex  factors.  Solving  equations  by  factoring.  Since 
(a  +  bi)  {a  —  bi)  =  r/  +  b^,  wherein  a  and  b  may  be  any  real  num- 
bers whatever,  therefore  tlie  sum  of  any  two  real  positive  numbers 
may  be  separated  into  two  imaginary  factors. 

E.g.,x'^-\-^  =  {x  +  2i)'{x-2i);  aH- 3  =(a  +  iV3)(a- iVi) ;  a;2  +  2a:  +  5 
=  (a;  +  l)24-4  =  (a;  +  l  +  2z)(a;  +  l-2?);  x^-x'^  +  \  =  x^-2x^-\-l  +  x^ 
=  (a;2- 1)2  +  a;2  =  (a.2_  1  +  a; .  i)  (a:2-  1  -  X  .  0- 

Note.  Observe  that  the  most  important  step  in  the  above  factoring  is  first  to 
write  the  given  expression  as  the  sum  of  two  squares;  the  plan  for  doiug  this  is 
precisely  that  which  is  followed  in  §  70, 

The  following  examples  will  illustrate  the  use  of  imaginary 
factors  in  solving  certain  kinds  of  equations ;  this  method  will  be 
more  fully  treated,  however,  in  Chapter  XV. 

Ex.  1.    Solve  the  equation  a;^  +  2  a:  +  5  =  0. 

Solution.    Since  this  equation  may  be  written  in  the  following  forms  : 

22  +  2  a:  +  1  +  4  =  0, 

(x  +  1)2  +  4  =  0, 

(x  +  l+20(x  + 1-20=0, 

therefore  it  is  clear  (§  72)  that  the  only  values  of  x  that  satisfy  it  are 

those  that  make 

a:  +  1  +  2  i  =  0  or  x  +  1  -  2  i  =  0 ; 

i.e.,  the  given  equation  is  satisfied  if,  and  only  if, 

x  =  -l  —2i  or  x  =  -  1  +  2  I ; 

i.e.,  the  roots  of  that  equation  are   —  1  —  2  z  and  —  1  +  2  t. 

Ex.  2.    Solve  the  equation  x^  =  ^x  —  22. 

Solution.     This  equation  may  be  written  in  the  following  forms : 

a;2  _  4  ^  +  22  =  0, 

(a:  _  2)2  +  18  =  0, 

{x-2  +  i  vT8)(x  -  2  -  i  VT8)  =  0  ; 

hence  its  roots  are  2-ivl8  and  2  +  iVl8,  i.e.,  2-3V^  and  2  +  3a/^. 


262  ELEMENTARY  ALGEBRA  [Ch.  XIV 

EXERCISES 

3.  By  actual  substitution  verify  the  correctness  of  the  roots  found  in 
Exs.  1  and  2  on  page  251. 

4.  What  must  be  added  to  x^  —  8  x  in  order  that  the  sum  shall  be 
the  square  of  a  binomial  ? 

5.  Write  a:^  —  8  a;  +  25  as  the  sum  of  two  squares. 

Solve  the  following  equations  and  verify  the  correctness  of  your  results : 

6.  a:2  4- 25  =  8  a;.  8.   a;2  -  x  +  1  =  0.         10.   3x2 -5  a: +21  =  0. 
1.  x^  +  x-\-l=0.  9.   4  x2  +  9  =  0.  11.   x-*  +  a2x2  +  a*  =  0. 

12.  Write  an  equation  whose  roots  are  1,  i,  and  —  i  (see  §  72,  note). 

13.  Write  an  equation  whose  roots  are  1,  —\+li  V3,  and  —  \  —  \  i  V3. 

14.  If  s  =  -  ^  +  i  i  V3,  show  by  substitution  that  s^  +  s  +  1  =  0. 
What  other  root  has  this  equation? 

III.     FRACTIONAL  EXPONENTS 

153.  Fractional  exponents.*  In  §  137  it  is  shown  that  the 
exponent  of  the  radicand  and  the  index  of  the  root  may  both  be 
multiplied  by  any  integer,  or  both  be  divided  by  any  factor  which 
they  may  have  in  common,  without  changing  the  value  of  the 
expression.  This  property  at  once  suggests  that  these  numbers 
may  bear  to  each  other  relations  similar  to  those  of  the  numerator 
and  denominator  of  a  fraction. 

For  this  and  other  reasons,  some  of  which  will  presently  appear, 
it  is  customary  to  employ,  when  it  is  desired  to  indicate  that  roots 
are  to  be  extracted,  not  only  the  radical  sign,  the  use  of  Avhich  has 
already  been  explained,  but  also  what  is  known  as  a  fractional 
exponent.  This  new  symbol  may  perhaps  be  best  defined  by 
the  identity  p  _ 

p 
i.e.,  the  symbol  A*"  means  the  pth  power  of  the  rth  root  of  A,  and 
r  must  therefore  necessarily  represent  a  positive  integer,  while  p 
may  be  positive  or  negative. 

E.g.,  9^  =  (V9)5  =  35  =  243,  and  8^3"  =  ( v/8)-4  =  2-4  =  ^  =  -1. 

24      16 

•For  a  similar  treatment  of  fractional  exponents  see  Tannery's  Arithme'tique. 


152-153]  FRACTIONAL  EXPONENTS  253 

p 

The  expression  A'',  whatever  the  value  of  A,  is  usually  spoken 
of  as  a  fractional  power  of  A,  just  as  A^  is  called  a  positive  integral 
power,  and  A~^  a  negative  integral  power. 

In  the  next  few  articles  it  is  shown  how  to  use  this  new  symbol 
in  the  various  algebraic  operations  j  these  uses  will  further  justify 
its  adoption. 

For  the  sake  of  simplicity,  here,  as  in  §§133-145,  only  the 
principal  roots  (§  132)  are  considered,  and  for  these  roots  it  has 
already  been  shown  that  (VZ)^  =  V^  [§  134,  Eq.  (1)] ;  hence,  in 

p 

the  following  proofs,  either  (^/Ay  or  -y/A^  may  be  used  for  A"". 
p  p 

Note.  Although  '•,  in  the  expression  A>' ,  is  called  a  fractional  exponent,  and 
is  written  in  the  form  of  a  fraction,  and  although  it  will  presently  appear  that 
such  exponents  may  often  he  dealt  with  as  though  they  were  really  fractions, 
yet  it  must  he  carefully  remembered  that  they  are  not  fractions  at  a,\\;  this 
fractional-exponent  notation  is  merely  another  loay  of  indicating  that  roots  are 
to  be  extracted. 

EXERCISES 

1.  What  is  meant  by  the  symbol  -  ?  Has  it  the  same  meaning  when 
used  as  an  exponent? 

2.  Is  the  exponent  — ,  in  the  symbol  x^,  really  a  fraction  ?     What  is 

the  precise  meaning  of  a."  • 

3.  Is  it  correct  to  say  that  the  symbol  x«  is  merely  a  convenient  way 
of  indicating  the  mth  power  of  the  nth  root  of  a;?  Is  this  the  same  as 
the  nth  root  of  the  rnth  power  of  x,  when  only  the  principal  roots  are 
under  consideration? 

Express  each  of  the  following  radicals  by  means  of  the  fractional- 
exponent  notation: 


4.  W.  6.    V^.  8.    V{a+2xy.     10.    </a-%\ 

5.  {Vmy.  7.    y/¥^\         9.   3  62.^^^^.  11.    ^2aP{x  +  ^yy. 

Find  the  numerical  value  of  each  of  the  following  expressions,  and 
explain  your  work: 

12.  4I        14.   25"k  16.   3.32"l      18.    (^^°)^- 

13.  9I        15.   4.4"^.9i        17.   (.09)"^.       19.  169^  •  f^V^  .  isi 

*  First  write  —  for  — -. 
2  2 


254  ELEMENTARY  ALGEBRA  [Ch.  XIV 

Translate  the  following  into  equivalent  radical  expressions : 
20.   a».  22.   5.(^y  +  2{ax)K  ^4    3z^-7aW 


-oA-l  '  «"^  +  ^^^ 


21.   a^  +  68.  23.    -  2  a%  K 

25.  Of  the  following  expressions,  which  are  integral  and  which  are 
fractional  powers  (see  §153)?  Which  are  positive  and  which  negative 
powers?     Give  the  reason  for  your  answer  in  each  case. 

154.   Fractional  exponents   changed  to  lower  and  higher  terms. 

Under  the  above  definition  of  a  fractional  exponent  it  is  easily 

verified  that  16^  =  16^,  [Each  member  being  4 

and  that  9^  =    9'^.  [Each  member  being  27 

So,  too,  in  general,  if  A  is  any  numher  whatever,*  and  if 
—  is  any  simple  fraction  in  which  r  is  positive,  then 

p  pm 

A^=A:-f, 

wherein  m  is  any  positive  integer  whatever. 

The  proof  of  this  statement  follows  directly  from  the  definition 
of  a  fractional  exponent  and  from  §  137,  for 

A-  =  </AP  [§153 

='V:^  [§  137 

=  Jr^,  [§  153 

p      pm 

i.e.,  A''  =  A''"",     which  was  to  be  proved. 

*If  9'  is  even  A  must  be  positive,  since  imaginaries  are  exchided  from  this  dis- 
cussion (cf.  also  footnote,  p.  2'2<J). 

t  Observe  that  this  equality  can  not  be  affirmed  merely  because  —  =  — ,  con- 
sidered as  fractions. 


153-155]  FRACTIONAL  EXPONENTS  255 

Note.  Observe  that  the  proof  of  §  154  applies  to  real  numbers  only ;  if  imagi- 
nary numbers  present  themselves,  here  or  elsewhere,  they  must  be  dealt  with  in 
accordance  with  the  principles  given  in  §§  146-152, 

Ex.  1.  By  means  of  fractional  exponents  reduce  y/a^  and  Vx^  to 
equivalent  radicals  of  the  same  order.  ' 

Solution.  The  given  radicals  are  respectively  equivalent  to  aJ  and 
a:^,  and  these  expressions  are  respectively  equivalent  to  as  and  a;^s\  i.e.,  to 
y/a^  and  y/x^^,  each  of  which  is  of  order  6. 


EXERCISES 

2.  Can  a^  and  x^  be  reduced  to  equivalent  expressions  whose  common 
order  is  any  multiple  whatever  of  2  and  3?    How? 

3.  State  in  detail  how  the  principle  proved  in  §  154  may  be  employed 
to  reduce  any  two  or  more  given  radicals  (real  numbers)  to  equivalent 
radicals  of  a  common  order. 

4.  Solve  Exs.  1-8  of  §  138  by  means  of  fractional  exponents. 

155.   Product  of  fractional  powers  of  any  number.     If  A  is  any 

nuiivber  whatever  (cf.  footnote,   p.   254),  and  if  ^  and  — 

r  r' 

are  any  two  simple  fractions  in  which  r  and  r'  are  positive, 

then 

p         p'  pr'+p'r 


A-'A'-  =  A  ^'^  . 

For,  since 

A'-=A'-''=s/A^'^, 

[§§  154  and  153 

and  since 

p 
A- 

A'-=A'"-=Va^''', 
'A'-=''y/A^'^'''VA^-'- 

[§§  154  and  153 

therefore 

^'•^^4pr'+pV                [-§  133 

pr'+p-r 

=A  '■'•'  ,              [§  153 

which  was  to  be  proved. 

'     1_       '  .  I  pr'+p'r 

Since  £LslJLL  —  P^P^  -^e  may  write,  instead  of  A  ""  ,  the 
rr  r      r 


p^p 


simpler  form  A''  '"',  if  we  are  careful  to  remember  that  the  symbol 
A'  *"  is  to  be  interpreted  by  first  adding  the  exponents  as  though 


256  ELEMENTARY  ALGEBRA  [Ch.  XIV 

they  really  were  fractions.  With  this  understanding  the  principle 
which  has  just  been  proved  becomes 

p      p'        p.p^ 
A^'A'-'  =  A'-  ^ 
p      p^      p-         p  p^      p^        p+Pl^^ 
Similarly,  AT •  A''' •  A'"'  =  A'  ' ' •  A''  =  A^  ''  '^', 

and  so  on  for  any  number  of  factors ;  hence,  under  the  above 
definitions,  fractional  exponents  conform  to  the  exponent 
law 

Am  ,     An  ,     Ap  . . .   ^:^  j^m+n+p+-" 

already  demonstrated  when  m,  n,  p,  •••  are  integers. 

EXERCISES 

1.  What  is  the  numerical  value  of  let .  16^  -16^   Of  let+l+h 
Is  then  let .  lel .  16^  equal  to  16^+1+1? 

2.  Do  the  fractional  exponents  in  Ex.  1  conform  to  the  same  law  as  if 
they  were  positive  integers?    State  that  law. 

Without  extracting  any  irrational  roots,  reduce  the  following  expres- 
sions to  their  simplest  forms : 

3.  8t.8l.8i  5.   241.24^.24-1.  7.   d^.a^.ai. 

4.  8t.8t.8-i.        6.   5^  .  5"?  .  5t  •  55  •  5-tV         8.   2x^ -Zxi -^-^x^. 

1121     _8  21  ^-*?.^?i 

9.   a^b'^x^  'b'^x^^  .  a^b^.  10.  a'^arV  •  ar^z/*"  .  aTy"". 

11.  Show  that  every  step  in  the  proof  of  the  above  principle  (§  155) 
remains  valid  even  if  p'  should  be  negative  (cf.  Ex.  4) ;  and  also  if  jo  =  r, 
or  if  there  is  any  other  relation  among  p,  r,p',  and  /. 

156.  Quotient  of  fractional  powers  of  any  number.  From  the 
definition  of  a  fractional  exponent  (§  153)  it  follows  directly  that 

64^  ^  64^  =  64^,  i.e.,  64^"^,  [Each  being  2 

and  that        64^  h-  64^  =  64"^,  i.e.,  64^.  [Each  being  | 

So,  too,  in  general,  if  A  is  any  nwrnber  whatever  (cf.  foot- 


155-157]  FRACTIONAL  EXPONENTS,  257 

P  P' 

note,  p.  254),  and  if  —  and  -j  are  any  two  simple  fractions 

in  which  r  and  r'  are  positive,  then 


p 
A- 

r'               P    P'                                                             P    p-               pr'-pr- 

■r  A''  =  A"-  '■'.                [Where  A"-  '"'  =  A  - 

For,  since 

A^  =  A^-  =  yA^\              [§§  154  and  153 

and  since 

p 
A'- 

A'  =  A^-  =  VA^',              [§§  154  and  153 
p' 

therefore 

r-A^  =  Va^'--  ^  VA^-^  ='\/A^^--p-^        [§  135 

i.e., 

p 

pr'-p'r 

=  A  ^-^  .          [§  153 

p'             ppr 
■:-  A"  =  A'-    '•', 

which  was  to  be  proved.     This  proof  shows  that,  under  the  above 
definitions,  fractional  exponents  also  conform  to  the  law 

A""  ^  ^"  =  J'"-" 

already  demonstrated  when  m  and  n  are  integers. 

EXERCISES 

1.  What  is  the  numerical  value  of  16?  -  leh    Of  16?"^?    Is,  then, 
16?  -  16^  equal  to  IgH? 

2.  Do  the  fractional  exponents  in  Ex.  1  conform  to  the  same  law 
as  though  they  were  positive  integers  ?     State  that  law. 

Simplify  the  following  expressions: 

3.  St-si  5.    64t -f- 64t .  64i  7.   2a:t-4A 

4.  8t.8t-8i  6.    12t.l2^-12.  8.   x^  -  3  a^xi 
9.   Show  that  every  step  of  the  proof  of  the  above  principle  (§  156) 

remains  valid  even  if  />  =  0,  and  thus  prove  that  1  -f-  a"  =  a  «.     Com- 
pare this  result  with  §  44. 

10.  By  means  of  Ex,  9,  show  that  a' factor  may  be  transferred  from 
numerator  to  denominator,  or  vice  versa,  by  merely  reversing  the  sign 
of  its  exponent,  even  when  the  exponent  is  fractional  (cf.  Exs.  22-26,  §  93). 

157.  Product  of  like  powers  of  different  numbers.  From  §  153 
it  follows  directly  that 

8^  .  27^  =  (8  .  27)^,  i.e.,  2161        [Each  being  36 


258  ELEMENTARY  ALGEBRA  [Ch.  XIV 


So,  too,  in  general,  if  A  and  B  are  any  two  numbers  what- 

'er  (cf.  footnote,  p.  254),  c 
which  r  is  positive,  then 


P 
ever  (cf.  footnote,  p.  254),  and  if  --  is  any  simple  fraction  in 


A^ .  B  =  {ABY. 

For,  since 

J-  =  V^  and  R  =  V^, 

p      p 

[§153 

therefore 

A'- .  R=</A'' .  -\/B^  =  ^'A^  .  & 

[§  133 

=  </XABf 

[§  121  (iii) 

=  (AB)S 

[§153 

I.e., 

p      p              p 

A'- '  jr  =  {ABy, 

which  was  to  be  proved.     This  proof  shows  that,  under  the  above 
definitions,  fractional  exponents  also  conforin  to  the  law 

A''-B^  =  {ABY 

already  demonstrated  when  7i  is  an  integer. 

Moreover,  by  successive  applications  of  the  above  proof  it  fol- 
lows that 

p      p       p  p 

A^  .B-  .C'...  =  {ABC--)% 

for  any  number  of  factors  whatever 

EXERCISES 

1.  What  is  the  numerical  value  of  16^  •  9^?    Of  144*,  i.e.,  of  (16  •  9)^? 
Is,  then,  16^  •  9^. equal  to  (16  •  9)^? 

2.  Does  the  fractional  exponent  in  Ex.  1  conform  to  the  same  law  as 
though  it  were  a  positive  integer?     State  that  law. 

3.  Does  the  law  asked  for  in  Ex.  2  apph'-  to  products  of  three  or  more 
factors  as  well  as  to  products  of  only  two  factors?    Verify  it  for  the 

m  m  m 

product  st .  125^  •  .064^ ;   and  prove  it  for  o".s»x'». 

158.   A  power  of  a  power  of  a  number.     From  §  153  it  follows 
directly  that 

(64^)^  =  64%  i.e.y  64^  *  i  [Each  being  4 


157-159]  FRACTIONAL   EXPONENTS  259 

So,  too,  in  general,  if  A  is  any  numher  whatever  (cf.  foot- 

p  p' 

note,  p.  254),  and  if  —  and  77  are  any  two  simple  fra/ytions 

in  which  r  and  r'  are  positive,  then 

(  ?y       p.  £i 

Uv'-'  =  AT  '^ 

p         

For,  since  A'  =</A^,  [§  153 

therefore         u9^"  =  V(V^  =  V  V^  [§  134 

=VAP-  [§  136 

pp' 
=  A'-'-;  [§  153 


I.e.,  V^'7'-  =  yl'-'^ 


iJP 


which  was  to  be  proved.     This  proof  shows  that,  under  the  above 
definitions,  fractional  exponents  also  conform  to  the  law 

already  demonstrated  when  m  and  n  are  integers. 

EXERCISES 

1.  What  is  the  numerical  value  of  (729^)2?  Of  729^  Is,  then, 
(729^)^  equal  to  729^"^?     Is  it  also  equal  to  (729^)^? 

2.  Do  the  fractional  exponents  in  Ex.  1  conform  to  the  same  law  as 
though  they  were  positive  integers  ?     State  that  law. 

/    r\p  rp 

3.  Read  the  equation  ^x^^i  =  x'l ;  state  what  the  several  indicated 
operations  are ;  mention  the  order  in  which  they  are  to  be  performed ; 
and  prove  the  correctness  of  the  equation. 

159.  Summary  of  exponent  laws.  As  originally  used,  the  symbol 
A"  was  merely  an  abbreviation  for  the  product  A'  A-  A-'-to  n 
factors  [cf.  §  7  (iv)  and  also  §  37],  and  n  was  therefore  necessarily 
a  positive  integer.  Later  on  (§  44  )  it  was  found  desirable  slightly 
to  extend  the  meaning  of  an  exponent,  and  it  was  agreed  that 
A^  should  mean  1,  and  that  ^~*,  where  fc  is  a  positive  integer, 
should  mean  — .    Under  these  luterpretations,  it  was  then  proved 


260  ELEMENTARY  ALGEBRA  [Ch.  XIV 

(§  121)  that  when  m  and  n  represent  any  positive  or  negative 
integers  whatever,  including  zero,  then 

I                                           A"''A^  =  A"'+%  (1) 

{A-r  =  A-%  (2) 

A^'B^  =  (ABy,  (3) 

and                                    ^'^  -J-  ^"  =  A"'-^  (4) 

These  formulas  state  the  so-called  "exponent  laws."  It  has 
now  been  shown  (§§  155-158)  that,  under  the  definition  given  in 
§  153,  these  exponent  laws  remain  valid  even  when  some  or 
all  of  the  exponents  are  simple  fractions  (cf.  §  154,  note). 

EXERCISES 

1.  Translate  the  first  exponent   law  into   a   rule    for  multiplying 
together  two  different  powers  of  any  given  number. 

Find  tlie  following  products  and  explain  each;  does  the  rule  given  in 
Ex.  1  apply  in  finding  these  products  ? 

2.  53  .  54.  4.  (1)4  .  (1)2.      6.   83- .  sf.  8.   26"?  •  26-*. 

3.  126.12-4.        5.  64.6°.  7.   14^  •  14"i      9.   .Oil  •  .04"*. 

10.  State  in  detail  the  precise  meaning  that  we  have  agreed  to  give 
to  each  of  the  different  kinds  of  exponents  used  in  Exs.  2-9,  z'.e.,  the 
meaning  of  5^,  12-4,  go,  14^,  and  26-t. 

11.  State  briefly  the  important  steps  by  which  law  (1)  was  estab- 
lished when  m  and  n  are  positive  integers ;  when  one  or  both  are  negative 
integers ;  and  when  they  are  simple  fractions. 

12.  Prove  that  law  (1)  applies  also  to  product?  of  three  or  more 
powers  of  any  given  number,  —  e.g.^  that  a;"*  •  x"  •  a:'"  =  a;'»+'»+'*,  where  m,  n, 
and  r  may  be  integers,  fractions,  or  zeros. 

9  2     5 

13.  Translate  law  (2)  into  a  rule  and  employ  it  to  simplify  (8^)^. 

14.  Make  up  3  examples  to  illustrate  the  application  of  law  (2)  with 
the  various  kinds  of  exponents  (cf.  Exs.  2-9  above). 

15.  Is  {ix'^Y'Y  equal  to  a;"»'»'?  Why?  May  m,  n,  and  r  be  fractions 
as  well  as  integers  here ?    May  one  or  more  of  them  be  negative?    Zero? 

16.  Show  that  (a-2)-3  =  ^1  ^  "^  =  ^^  =  i  =  a^.  Is  this  the  same  as 
a(-2).(-3)?    [Cf.§121(ii)].   "  (-2)      -6 

17.  As  in  the  first  part  of  Ex.  16  show  that  (m~^)~^  =  w?. 


159-160]  FRACTIONAL   EXPONENTS  261 

18.  Translate  law  (3)  into  a  rale,  and  state  what  limitations,  if  any, 
are  placed  upon  the  value  of  n. 

19.  Prove  that  law  (3)  applies  also  to  the  product  of  three  or  more 
like  powers,  —  i.e.,  that  a"*  •/>"»•  e"*  •  ^"^  •••  =  {abed  •••)'",  wherein  m  may  be 
positive  or  negative,  integral  or  fractional,  or  zero. 

20.  Make  up  4  examples  to  illustrate  the  application  of  law  (3)  with 
the  various  kinds  of  exponents. 

21.  Translate  law  (4)  into  a  rule,  and  illustrate  its  application. 

22.  What  is  the  product  of  ^'»-"  by  yl**?  What,  then,  is  the  quotient 
of  A'^  divided  by  A'^  [cf.  definition  of  division,  §  3  (iv)]  ? 

23.  By  means  of  the  suggestion  contained  in  Ex.  22,  prove  law  (4) 
from  law  (1*)  and  the  definition  of  division,  —  independent  of  §  156. 

160.   Operations  with  polynomials  involving  fractional  exponents. 

Since  the  operations  with  polynomials  are  merely  combinations  of 
the  corresponding  operations  with  monomials,  therefore  the  prin- 
ciples already  demonstrated  (§§  155-159)  for  monomials  suffice 
for  operations  with  polynomials  also. 

Moreover,  since  fractional  exponents  obey  the  familiar  laws 
formerly  established  for  integral  exponents,  and  since  any  radical 
expression  may  be  written  in  the  fractional-exponent  notation, 
therefore  operations  with  radicals  (real  numbers)  are  usually 
greatly  siraplilied  by  using  fractional  exponents ;  *  this  is  illus- 
trated below. 

Ex.  1.   Find  the  product  of  3  Va  —  5  v^    by    2-\/a-\-Vy. 

Solution.  Since  3Va  —  5\/^  =  3a^  —  5^/^,  and  2\/a -}- \/y  =  2a2  +  ^^> 
therefore  this  product  becomes 

3  a^  -  5  2^i 

6  ai+^  -  10  a^y^ 

+  3  o^y^  -  5  yi+^ 

6  a  —  7  a^y^  —  5  y^. 
If  it  is  desired,  this  product  may,  of  course,  be  written  in  either  of  the 
following  forms  :6a  —  7  Va  v^  —  by/y^    or    Q  a  —  7Va^y^  —  oVy^. 

*  Although  the  radical  notation  and  the  fractional-exponent  notation  are  each 
equivalent  to  the  other,  and  either  may  therefore  replace  the  other,  yet  each  is 
frequently  met  with,  and  it  is  desirable  that  the  student  should  understand  how 
to  operate  with  each  form  without  first  converting  it  into  the  other. 


262 


ELEMENTARY  ALGEBRA 


[Ch.  XIV 


Ex.  2.   Divide  x^  —  ij^  by  \^x  +  y/y. 

Solution.     Since  Vx  +  Vy  =  x^  +  y-,  this  solution  may  be  put  into 


the  follow ino-  form 


x^  +  x'^y"^ 


1         1 
x^  +  y^ 


5  4     1  2     3 

x3  —  a^3y2  ^  xy  —  xay^  + 


5     1  4. 

—  x^ys^  —  x"Sy 


xsy  —  y^ 


x^i 


-^W' 


xy^ 


x^y^ 


xsy^  -  2/3 
x'^t/^  +  x^y^ 

-y' 

The  above  quotient  may  also  be  written  thus  : 

Vx^  —  y/x"^  Vy  -{-  xy  —  Vx'^  Vy^  +  v^a:  •  y^  —  Vp. 

Note.  To  appreciate  one  of  the  advantages  of  fractional  exponents  the  student 
has  only  to  perform  the  division  in  Ex.  2,  using  the  radical  notation,  and  compare 
his  work  with  the  above  solution. 

Ex.  3.     Extract  the  square  root  of  v^  —  2  y/x^  +  5  v^x^  —  4  aXx  +  4. 

SoLUTiox.  This  expression  written  in  the  equivalent  fractional-expo- 
nent form  is  X*  —  2  z^  -1-  5  a;5  —  4  x"^  +  4,  and  in  this  form  its  square  root 
may  be  extracted  just  as  though  it  were  a  rational  expression  (cf.  §  125)  ; 

thus:  4  s  9.  1  9.         ^ 

x^  -  2  x"5  -I-  5  x^  -  4  x^  -f  4  [x?  -  x^-f  2 

4 
X^ 


2  x3  -  x^ 

-  2  x3  +  5  xt 

-2x^  +  xi 

2  x^  -  2  x3  + 

2 

4^!-4x*  +  4 
4x^-4x3  +  4 

hence   the   square   root    of    the   given    expression    is   xs  —  x3  -f  2,  i.e., 
^^-Vx-\-  2. 


160]  FRACTIONAL   EXPONENTS  263 

EXERCISES 

Perform  the  following  multiplications : 

4.  a^  +  62  by  a2  -  h^  (cf.  §  58). 

5.  a;3  —  x'^y'^  +  ?/3  by  :c3  -|-  yz, 

6.  771^  —  w"5n5  +  nS  by  //??  +  n^. 

7.  m^  —  m^y~^  +  n~a  by  w?^  +  n~o. 

8.  i  a:t  -  Jj  x?/^  +  ,V  ^^^  -  2V  2/^  by  ^  a:i  +  1  yi. 

9.  81  ^7^-27  ^^3^^+9^2^^_3^^^^^^  by  3  ^  +  ^y. 

10.  Va  -  4  VlK  +  6  \/^-  4  v'^  -4-  Vx  by   v^a  -  2  v^  +  Vx. 

11.  \/x^  +  2  Vp  -  \/2^  -  ^  Vy  +  2  V^  v^s  -  ^^  Vz 

by  y/x  —  2  Vy  +  Vs. 

12.  w^  +  m"t  -  2  m^  +  4  m~^  by  1  +  2  'm"^  -  -j^- 

13.  j9"t  +  ^-1-6  -  p-'!5q-^  by  />- -75  +  9-5. 

14.  14  n^x  a/x  +  2  n  Vn  +  1  a:^-^  +  6  n  v^  by  Vn  —  3  x*  +  7— -x^. 

15.  5  a-H-i  +  3  a-26»x-i  -  h^-^x^  by  x-^  -  3  ir~h-^  +  ai 

Perform  the  following  divisions  : 

16.  a  +  x2  by  a^  +  x^. 

17.  m5'  —  n3  by  m^  —  ns. 

18.  x-i  +  3  y~^  -  10  x?/-!  by  x"!  Vy  -  2. 

19.  a*  +  2  v^H  +  ^  by   v^  +  ft'i 

20.  x^  +  x'^y/y  —  xy/xy^  —  xy  +  -\/x  y^  +  y^  3  by  Vx  +  v^y. 

Simplify  the  following  expressions : 

21.  (^^  +  '^1'^  _  ^-Vy^         23  _^^ 9^ ]_<       1 


\Vx-y/'y'     \^x  +  Vy  y/a-1      y/a  +  1      «i-l      a^  +  1 

22       3:"»  +  ?/"     _     x"  -  y"^  24.  ^          ^       1 

'x— +  2/-«     x-^-2/—  '  y+Vy-^l'  yl-l 

25       ^  ~  y     _  Va:^  —  y'^ 
Vx  —  Vv        ^  —  y 


264  ELEMENTARY  ALGEBRA  [Ch.  XIV 

Extract  the  square  root  of  each  of  the  following  expressions : 

26.  a;2  +  2  a:t  +  3  a;  +  4  a:^  +  3  +  2  a;"i  +  x'K* 

27.  tt^  -  4  as  4-  4  «  +  2  a^  -  4  a^  +  ai 

28.  ns  —  2  nT^n'^'  +  2  rri^n^  +  m~^n  ^  —  2  m^n^  +  m^. 

Write  down,  by  inspection  if  possible,  the  square  root  of  each  of  the 
following  expressions : 

29.  1-2  m3  +  wi  31.  jo^  -  4  +  4;>~i 

30.  x^  4-  4  xt  +  4.  32.   axt  +  2  a^x^  +  atx. 

33.   m  +  n+jo  —  2  ni^n^  +  2  n^jo^  -  2  m^jo^. 

Extract  the  cube  root  of  each  of  the  following  expressions ;  write 
the  results  first  with  all  the  exponents  positive,  and  then  replace  all 
fractional-exponent  forms  by  radical  signs : 

34.  8  +  12  xt  +  6  art  +  x\ 

35.  8  x-i  -  12  x~iy  +  6  x'^nf  -  if. 

36.  r^  -  6  ri  +  15  ri  -  20  +  15  f^  -  6  f  +  it. 

37.  8  asrt  +  9  a&*  +  13  at  +  3  a^6  +  18  a%-^  +  &t  +  12  ah-K 

161.  Rationalizing  factors  of  binomial  surds.  Another  advantage 
of  the  fractional-exponent  notation  is  that  it  furnishes  an  easy 
method  for  finding  a  rationalizing  factor  of  any  binomial  surd 
whatever,  —  only  quadratic  binomial  surds  have  thus  far  been 
rationalized  (§  144). 

To  illustrate  this  method,  let  it  be  required  to  rationalize  the 
binomial  surd  cc^  _j_  y^^ 

Since  (xi)"—  (?/^)"  is  exactly  divisible  by  xs  +  y'2  whenever  n  is  an  even  posi- 
tive integer  [§  68  (ii)],  therefore,  if  n  be  given  such  an  even  integral  value  as  will 

make  both  (xs)**  and  (r/i)**  rational,  — e.g.,  G,  12,  18,  — ,  — then  the  quotient  of 

(xi)**  —  (?/^)'*  divided  by  x'S  -fy^  will  be  a  rationalizing  factor  of  a;3  +  y^,  because 

the  product  of  x'S  +  ?/^  by  this  quotient  will  be  (a;:?)"  —  (,?/2)",  which  is  rational  for 
all  such  values  of  n. 

*  Observe  that  this  expression  is  arranged  according  to  descending  powers  of  x. 


160-161]  FRACTIONAL  EXPONENTS  265 

In  the  present  case,  G  is  the  smallest  admissible  value  of  n,  and  the  required 
rationalizing  factor  is 

(a;3)6  —  (j/i)6      a;2  _  yS         5        41,  231  5 

«3  +,?/2  K^  +  y^ 

Again,  a  rationalizing  factor  of  x's  +  y'5  is  the  quotient  \.{x^)^^ -\- {y^Y^'\-^ 

(a;7  +  y'S),  i.e.,  {x^  +  y^)  -^  (k^  +  y's) ;  and  a  rationalizing  factor  of  a^  —  6*  is  the 

quotient  [(at)i2-  (6l)i2]  ^  («!  _  5!)^  i.e.,  (aS-  69)  -^  (al-  6^). 

The  student  may  now,  from  the  above  examples,  formulate  a 
rule  for  finding  a  rationalizing  factor  for  any  binomial  surd;  he 
should  distinguish  three  cases,  viz.,  (1)  when  the  binomial  is  a 
difference;  (2)  when  it  is  a  sum  and  the  L.  C.  M.  of  the  denomi- 
nators of  its  fractional  exponents  is  odd;  and  (3)  when  it  is  a 
sum  and  this  L.  C.  M.  is  even. 

EXERCISES 

Find  the  simplest  rationalizing  factor  for  each  of  the  following 
expressions : 

1.  a?  -  ri     2.  mk  +  ni     3.  2  a:l  -  3  ^i     4.  ahi  +  3  v\     5.  x"^  +  2  yl. 


CHAPTER   XV 

QUADRATIC   EQUATIONS 

I.     EQUATIONS   CONTAINING  BUT  ONE  UNKNOWN  NUMBER 

162.  Introductory  remarks.  It  has  already  been  shown  that  the 
first  step  in  solving  an  algebraic  problem  is  to  translate  its  condi- 
tions into  algebraic  language,  and  also  that  this  translation  leads 
to  equations  which  contain  one  or  more  unknown  numbers ;  the 
values  of  these  unknown  numbers  are  then  found  by  solving  the 
equations  (§  26). 

Although  nearly  all  of  the  problems  thus  far  met  with  are 
such  that  their  conditions  give  rise  to  equations  of  the  first 
degree  in  the  letters  representing  the  unknown  numbers,*  yet 
there  are  many  other  problems  which  lead  to  equations  of  the 
second  degree  in  those  letters;  the  solution  of  equations  of  this 
kind  will  be  investigated  in  the  present  chapter. 

Note.  It  may  be  recalled,  however,  that  some  easy  equations  of  the  second 
degree  have  already  been  solved  by  means  of  factoring  (§  72) ;  it  will  presently 
appear  that  all  such  equations  may  be  solved  by  the  same  method. 

163.  Definitions.  An  integral  algebraic  equation  which  involves 
the  second  but  no  higher  degree  of  a  number,  is  called  a  quadratic 
equation  in  that  number  (cf.  §  94). 

E.g.,  a;2 -f- 5  =  0,  'ix'^  —  ^  =  lx,  and  ax^  +  6a;  +  c  =  0  are  quadratic  equations  in 
the  number  represented  by  a;;  4c2  +  2c  =  9  and  a  (c  +  4)2  —  3  c  +  8  =  0  are  quad- 
ratic equations  in  c ;  and  a  (?/  —  3)2  -f-  6  (?/  —  3)  —  6  =  0  is  a  quadratic  equation  in 
y  —  3,  and  also  in  y. 

Unless  the  contrary  is  expressly  stated,  a  quadratic  equation  is  understood  to 
mean  a  quadratic  equation  in  the  unknown  number. 

Every  quadratic  equation  in  one  unknown  number,  say  x,  may 

evidently,  by  transposing   and   simplifying,    be  reduced  to  the 

standard  form  2.7,  a 

ax^  -+-  6a;  +  c  =  0, 

*  For  the  solution  of  first  degree  equations  see  Chapters  X  and  XI. 
266 


162-163]  QUADRATIC  EQUATIONS  267 

wherein  a,  b,  and  c  represent  known  numbers  and  are  usually 
called  the  coefficients  of  the  equation ;  the  term  free  from  x,  viz.,  c, 
is  also  called  the  absolute  term.  Although  6  or  c  may  be  zero,  a 
can  not  be  zero,  for  if  a  =  0'  the  equation  becomes  bx-\-  c  =  0, 
which  is  not  quadratic. 

If  neither  b  nor  c  is  zero,  the  equation  is  called  a  complete  quad- 
ratic equation,  while  if  either  6  or  c  is  zero,  it  is  called  an  incomplete 
quadratic  equation.  If  6  =  0,  the  equation  is  also  often  called  a 
pure  quadratic  equation,  otherwise  it  is  called  an  affected  quadratic 
equation. 

E.g.,  the  equation  2x^  +  5  —  3a;  =  7a;  —  8  becomes,  by  transposing  and  uniting 
terms,  2  a;2  _  lo  ^  -{- 13  =  o,  which  is  in  the  above  standard  form,  —  the  coefficients 
a,  b,  and  c  of  the  general  equation  being  for  this  particular  case  2,  — 10,  and  13, 
respectively;  it  is  a  complete,  and  also  an  affected,  quadratic  equation. 

Again,  the  equation  8x2  +  4  — 3  a;  =  ^^— — '-  —  x  +  S  becomes,  by  clearing  of 

fractions,  transposing  and  uniting  terms,  16x^  —  3  =  0,  which  is  in  the  standard 
form,  a,  b,  and  c  being  16,  0,  and  —  3,  respectively;  it  is  an  incomplete,  and  also 
a  pure,  quadratic  equation. 

In  the  same  way  evenj  quadratic  equation  in  one  unknown  number  may  be 
reduced  to  the  standard  form. 

EXEFiCISES 

1.  What  are  the  important  steps  in  the  solution  of  an  algebraic 
problem  (of.  §  26)  ?     What  is  meant  by  the  "  equation  of  a  problem  "? 

2.  If  the  conditions  of  a  problem,  when  translated  into  algebraic 
language,  lead  to  a  quadratic  equation  (such  as  5  a;^  —  8  x  +  10  =  0),  can 
that  problem  be  solved  by  the  methods  given  in  Chapter  III  or  Chap- 
ter X? 

3.  What  is  a  numerical  equation?  a  literal  equation?  a  simple 
equation?  a  general  equation?  a  particular  equation?  a  root  of  an 
equation  ? 

4.  Is  3  a:2  —  2  a:  =  0  a  complete  or  an  incomplete  quadratic  equation  ? 
Why  ?     Is  it  pure  or  affected  ?     Why  ? 

5.  Reduce  5x^  +  2  —  8x  =  4(8  —  x)  to  the  " standard  form."  What 
is  its  absolute  term?  Is  this  equation  pure  or  affected?  complete  or 
incomplete  ?     Why  ? 

6.  Clear  the  equation  2a:  —  3+-=a:  +  2of  fractions,  then  reduce  it 

to  the  standard  form,  and  classify  it  (pure,  complete,  etc.) ;  also  solve  it 
by  the  method  of  §  72. 


268  ELEMENTARY  ALGEBRA  [Ch.   XV 

7.  Is  the  equation  in  Ex.  6  a  quadratic  or  a  simple  equation  ?    Why  ? 

8.  If  X  andy  stand  for  unknown  numbers,  tell  which  of  the  following 
equations  are  simple,  which  quadratic,  and  which  of  a  still  higher  degree : 

a4ar2  +  a^x  -\-  a  =  0',     ^^=-^  =  -;    5  x  -  7  ?/  =  11 ;    5  x  +  ^'^  -  7  ^  =  11 ; 
2         X 

'2(x2-  x)+6  =  2x^;     ^  -  4:  =  ')  x  +  -^  ;    3  ar  +  4  a^  _  o  ax  =  7. 

y  y  +  - 

9.  What  particular  equation  is  obtained  by  substituting  the  values 
2,  —  7,  and  5  for  the  coefficients  in  the  general  equation  ax^  +  hx  +  c  =  01 

10.  By  assigning  different  sets  of  values  to  the  letters  a,  6,  and  c,  how 
many  particular  quadratic  equations  can  be  formed  from  the  general 
equation  ax^  +  ftx  +  c  =  0  ? 

Why  is  this  last  equation  called  a  "general"  equation,  and  one  in 
which  the  coefficients  are  numerals  a  "  particular  "  equation  ? 

164.  Solution  of  quadratic  equations.  Although  the  roots  of  any- 
quadratic  equation  whatever  may  be  found  by  the  method  of  fac- 
toring (§§  72  and  165),  yet  there  are  various  other  methods. for 
solving  these  equations,  and  one  of  these,  which  will  doubtless  be 
more  easily  followed  by  the  student,  will  now  be  explained. 

Ex.  1.     Find  the  roots  of  the  equation  2x2-3-5a:  =  7a:+ll. 

Solution.     By  transposing  and  uniting  terms,  the  given  equation 

becomes 

2x^-12x=,U,  (1) 

whence,  dividing  by  2,  x^  —  6  x  =  7 ;  (2) 

if  now  9  be  added  to  each  member  of  Eq.  (2),  it  becomes 

x^-Qx  +  9  =  lQ,  (3) 

i.e.  (see  "  remark "  below),        (z  -  3)2  =  16,  (4) 

whence,  taking  square  roots,         x  —  3  =  ±  4,  (5) 

i-e.y  a:  -  3  =  +  4,  or  X  -  3  =  -  4,  (6) 

hence,  transposing,  x  =  7,  or  x  =  —  1, 

and,  on  substituting  these  values  of  x  in  the  given  equation,  it  is  found 
that  they  each  satisfy  that  equation  ;  they  are,  therefore,  the  roots  of  the 
given  equation. 

That  this  equation  has  no  other  roots  is  shown  in  Ex.  38  below. 


163-164]  QUADRATIC  EQUATIONS  269 

Eemark.  Since  (x  ±  k)-  =  oi^.  ±  2  kx  +  Jc'y  therefore  the  expres- 
sion x^  ±  2kx,  whatever  the  value  of  A:,. lacks  only  the  term  k'^  of 
being  the  square  of  x  ±k,  i.e.,  if  the  square  of  half  the  coeffi- 
cient of  the  first  power  of  x  he  added  to  an  expression  of 
the  form  x^  +  hx,  the  result  will  he  an  exact  squared 

E.g.,  if  (  -^  )    be  added  to  a;2  —  (j  x,  the  expression  becomes  {x  —  3)2,  as  in  Eq.  (3) 

above ;  if  (  -  j    be  added  to  y'^  +  5  ?/,  it  becomes  (?/+-)   ;  and  if  (  7  )    be  added 
to  z'^+hx,  it  becomes  lx-\--\  • 

(Og   l^y,  \    2 

"  \  ,  i.e.,  72,  be  added  to  4  k'^x'^  +  28 kx,  it  becomes 

2V4/fc2j;2y 

(2  kx  +  7)2 ;   this  may  also  be  seen  by  first  writing  4  ^2^2  -f  28  kx  in  the  form 
(2A:x)2^-14(2^•x). 

Ex.  2.     Solve  the  equation  a;2+llx  +  l  =  8a;. 

Solution.     On  transposing,  the  given  equation  becomes 

x^  +  Sx  =  -l,  (1) 

whence,  adding  (|)2,        x^  -{- Sx +(1^  =  -  1  +  (|)2,  (2) 

i-e.,  (X  +  1)2  =  I,  (3) 

and  hence  a:  +  f  =  ±A/|=±^  \/5,  (4) 

.^-l±lvE  =  :^lf^,  (5) 

and  each  of  these  values  of  x,  viz.,  ~  '  "^ — '-  and  — — ^ — '-,  is  found,  on 

substitution,  to  satisfy  the  given  equation ;  they  are,  therefore,  the  roots 
of  that  equation. 

Ex.  3,     Solve  the  equation  ax^  -^  bx  +  c  =  0. 

Solution.     On  transposing  and  dividing  by  a,  this  equation  becomes 

x'^  +  ^x  =  -^',  (1) 

a  a 

whence  x^  +  -x  +  {  —      =  — — = — ,  (2) 

a         \2a/        4a2      a  ^a^  ^  ^ 


I.e., 


I         b  \2_ft2_4ac  ,3. 


*  Making  this  addition  to  the  given  expression  is  usually  spoken  of  as  com- 
pleting the  square. 


270  ELEMENTARY  ALGEBRA  [Ch.  XV 


therefore                                ,  ^  ^  =  ±  J^l^^  =  ±2^EZ±££,       (4) 
I.e.,  a;  =  -  — -  + = ,  (5) 


and  as  before,  each  of  these  values  of  x,  viz.,   '^^—— —  and 

2  a 

— — — — ^—^ — —,  is  a  root  of  the  given  equation. 
2a 

Note.  Having  now  shown  how  to  find  the  roots  of  any  quadratic  equation 
whatever,  the  method  of  §  67  may  be  employed  to  find  the  factors  of  any  quadratic 
expression  of  the  form  ax^  -\-hx-\-  c  (cf.  also  §  165). 

E.g.,  since  7  is  a  root  of  the  equation  x^  —  6  a;  —  7  =  0  (see  Ex.  1  above) ,  there- 
fore X  —  7  is  a  factor  of  the  expression  x^—iSx  —  1  (cf .  §  67) . 

Similarly,    from  Ex.   2,  the  factors  of    x^  +  Zx-\-\    are  x  —  ~^'^—   and 

z  —  — — ~        ;  and  x  —  ~    "*"         ~ — —  is  a  factor  of  the  expression  ax^+bx+c. 
2  2  d 


EXERCISES 

4.  In  Ex.  1  above,  how  was  Eq.  (1)  obtained  from  the  given  equation? 
State  also  how  Eq.  (2)  was  obtained  from  Eq.  (1)  ;  Eq.  (3)  from  Eq.  (2) ; 
Eq.  (5)  from  Eq.  (3).  How  many  equations  are  expressed  in  (5)  ?  How 
were  the  roots  of  the  given  equation  finally  found  from  Eq.  (5)  ? 

5.  Show  that  the  essential  steps  in  the  solution  of  Ex.  2,  and  of  Ex.  3, 
are  the  same  as  those  in  Ex.  1,  viz., 

(1)  Transposing  and  uniting  terms,  and  dividing  each  member  of  the  new 
equation  by  the  coefficient  of  the  second  power  of  the  unknown  number,  thus  re- 
ducing the  given  equation  to  the  form  x^  +  mx  =  n  ;  (2)  adding  (  —  )    to  each 

member,  thus  making  the  first  member  an  exact  square ;  (3)  extracting  the 
square  root  of  each  member  {giving  the  double  sign  to  the  second  member^, 
and  solving  the  two  resulting  simple  equations. 

By  the  above  method  find  the  roots  of  the  following  equations,  and 
verify  the  correctness  of  each : 

6.  2  a;2  -  27  =  9  X  -  a;2  +  3.  12.  5  a:  =  x2  -  14. 

7.  2;2+5.r  =  21  +  a:.  13.  19  a:  +  5a:2  =  15  -  5a:2. 

8.  2/2-52^-24  =  0.  14.  2y^-by=^y-\-2U. 

9.  2  a:2  -  a:  =  3.  15.  22  f  +  3  /2  =  4  ^2  _  43. 

10.  2.^2  -  10^/  =  ?/2  +  lOy  -  51.  16.   0  /2  _  3  =  10  ^  -  3<2. 

11.  z^^  z-  1.50  =  4  -  2 e.  17.   9  -  5 0,2  =  12 x. 


164]  qUADRATIC  EQUATIONS  271 

18.  Write  a  carefully  worded  rule  for  solving  such  equations  as  those 
given  above ;  also  show  that  by  this  rule  any  quadratic  equation  what- 
ever, which  contains  but  one  unknown  number,  may  be  solved. 

19.  Show  that  the  rule  asked  for  in  Ex.  18  will  serve  to  solve  such 
equations  as  x^  +  6  x  =  0.  What  are  the  two  roots  of  this  equation  ? 
Verify  your  answer. 

20.  Show  that  while  such  equations  as  that  given  in  Ex.  19  may  be 
solved  by  the  above  method,  they  may  be  much  more  easily  solved  by  the 
method  given  in  §  72. 

Prove  that  if  an  equation  has  no  absolute  term,  one  of  its  roots  is  neces- 
sarily zero. 

21.  Does  the  rule  asked  for  in  Ex.  18  apply  to  such  equations  as 
4  ^2  —  9  =  0  ?  What  are  the  roots  of  this  equation  ?  Verify  your 
answer. 

Solve  the  following  equations,  and  verify  your  results : 

22.  5  a;2  =  8  z.  25.   ax^  +  bx  =  cx^. 

23.  lSx'\-2x^=5x  +  4x^.  26.   ax^  +  b  =  0. 


24.   3f-8y  =  2y(y-i)-\-Q.  27.    {m  +  n)x'^  +  n^ 


m 


28.  What  must  be  added  to  a;^  +  8  a:  to  "  complete  the  square  "  ? 

29.  What  must  be  added  to  P^  —  5  P  to  complete  the  square  ? 

30.  What  must  be  added  to  (x  +  yy  —  4:(x  -h  y)  to  complete  the  square  ? 

31.  What  must  be  added  to  4:  M^  -\-  8  M  to  complete  the  square  ? 

32.  What  must  be  added  to  9  a'^x^  +  12  ax'^  to  complete  the  square? 

33.  Show  that  the  answer  to  each  of  the  exercises  28-32  conforms  to 
what  is  said  in  the  "remark"  under  Ex.  1. 

34.  How  many  different  equations  are  expressed  by  P  =  ±  Q  ?  What 
are  they  ?    Write  them  separately. 

35.  How  many  different  equations  are  expressed  by  ±  P  =  ±  Q  ?  What 
are  they?  Write  them  separately.  Do  the  equations  +  P  =  +  Q  and 
—  P  =  —  Q  express  the  same  or  different  relations  between  P  and  Q? 

36.  Show  that  the  equation  P  =  ±Q  expresses  all  the  relations  between 
P  and  Q  that  are  expressed  by  the  equation  ±P  =  ±  Q ;  and  hence  show 
that  the  double  sign  (±)  need  be  employed  in  only  one  member  of  an  equa- 
tion which  is  obtained  by  extracting  the  square  root  of  each  member  of 
a  given  equation.     Illustrate  this  in  the  solutions  of  Exs.  1  and  2  above. 


272  ELEMENTARY  ALGEBRA  [Ch.  XV 

37.  Prove  that  the  two  equations  P  =  ±  Q  are  together  equivalent 
(§  95)  to  the  equation  P^  =  Q^ 

Proof.    The  equation  P^  =  Q^ 

is  equivalent  to  pa  _  q2  =  o,  [§  95  (1) 

i.e.,  to  (P-Q)(i'+Q)  =  0, 

and,  manifestly,  this  last  equation  is  satisfied  when,  and  only  when, 

P  -  Q  =  0  or  P  +  Q  =  0, 
i.e.,  when  P  =  ±  Q ; 

hence  the  equations  P2  =  Q2  and  P  =  :kQ  are  equivalent. 

38.  In  the  solution  of  Ex.  1  above,  show  that  the  given  equation  and 
Eqs.  (1),  (2),  (3),  and  (4)  are  all  equivalent  to  each  other,  and  that  each 
is  equivalent  to  the  two  equations  (5),  i.e.,  to  the  two  in  (6).  Hence  show 
that  the  given  equation  has  two  roots,  and  only  two. 

39.  By  the  method  of  Ex.  38,  show  that  the  equation  given  in  Ex.  2, 
above,  has  two  roots,  and  only  two. 

40.  Show  that  Ex.  3  has  two  solutions,  and  only  two,  and  thus  prove 
that  every  quadratic  equation  in  one  unknown  number  has  two  roots,  and 
only  two  (cf.  §  97). 

Solve  the  following  equations,  and  verify  your  results  : 

41.  3  a;2  +  5  x  -  4  =  x2  -  2  a:  +  3.  45.  2y'^  +  Z  =  l  y. 

42.  x2-|x-2  =  0.  *6.   3x2-10  =  7x. 

47.   6  +  5  «  =  6  f2. 

43.  (2-x){x+\)+^  =  x-^.  ^ 

48    Ix  —  '^-^  4-  '^  =  0 

44.  (2  2/-3)2zz:6(?/  +  l)  -5.  ^  4 

49.  What  are  the  roots  of  x^  —  Sar  —  2  =  0?  Are  these  roots  rational 
or  irrational  numbers?  Define  rational  and  irrational  numbers.  Are 
the  above  roots  real  or  imaginary  ? 

50.  What  are  the  roots  of  a;^  —  3a:  +  4  =  0?  Verify  the  correctness 
of  your  answer.     Are  these  roots  real  or  imaginary  ? 

51.  Solve  the  equation  3a;2-8a:+10  =  0. 

Suggestion.  The  method  already  explained  for  solving  such  equations  gives 
rise  to  fractions ;  these  fractions  can  be  avoided  by  proceeding  thus : 

On  multiplying  the  given  equation  by  3  (the  coefficient  of  a;2),  and  transposing,  it 

^^°°"^««  9x2_24cc=-30; 

completing  the  square,  9  a;2  —  24  a;  + 16  =  —  30  +  16  =  —  14, 

i.e.,  (3  X -4)2  =—14, 

hence  3  «  —  4  =  =t  V^^HI, 

and  a;  =  i(4  +  \/^=n4)  or  i(4  -  V^^14). 


164]  QUADRATIC   EQUATIONS  273 

52.  Solve  the  equation  3a;2-5a;-2  =  0. 

Suggestion.    Multiply  this  equation  by  4  •  3  and  then  proceed  as  in  Ex.  51. 

53.  Solve  the  equation  ax^  ■}-  bx  -{■  c  =  0. 
Multiply  by  4  a  and  then  proceed  as  in  Ex.  51. 

54.  Solve  the  equation  mx'^  -\-  2  nx  -^  k  =  0. 
Multiply  by  m  and  proceed  as  in  Ex.  51. 

55.  By  studying  ^xs.  51-54,  especially  53  and  54,  point  out  when  it  is 
necessary  to  multiply  by  4  times  the  coefficient  of  the  second  degree  term 
in  order  to  avoid  fractions  in  the  solution  of  a  quadratic  equation ;  and 
also  when  multiplying  by  that  coefficient  alone  will  suffice. 

Solve  the  following  equations,  avoiding  fractions  in  completing  the 
square  : 

56.  3  a;2  +  2  X  =  7.  60.   2  t^  +  7  t  =  -  Q. 

57.  5  x2  +  6  a;  =  8.  61.   3  a.-2  -  5  x  =  2. 

58.  3y^  +  4y  =  95.  62.   5  z^  -  x  -  3  =  0. 

59.  2  !/2  +  3  3/  rr  27.  63.   15  y-^ -7  y  -  2  =  0. 

64.  Is  8  a  root  of  a:^  -  5  a:  -  24  =  0  ?  Why  ?  What  is  the  correspond- 
ing factor  of  a:2  —  5x  —  24  (cf.  Ex.  3,  note)  ?  What  is  the  other  factor 
of  this  quadratic  expression?  What  root  of  the  given  equation  corre- 
sponds to  this  other  factor  ? 

65.  Since  x'^  —  7  x  -\-  I0~(x  —  2)(x  —  o),  what  are  the  roots  of  the 
equation  a:2  -  7  x  +  10  =  0  ?     Why  (cf .  §  72)  ? 

66.  Since  2  and  7  are  the  roots  of  x^  —  9  x  +  14  =  0,  what  are  the 
factors  of  x2  -  9  X  +  14  ?     Why  (cf .  §  67)  ? 

67.  Since  ^  and  f  are  the  roots  of  6  x2  —  7  x  +  2  =  0,  what  are  the 
factors  of  6x2  —  7x  +  2?  ^^.g  these  the  only  factors,  or  is  there  also  a 
numerical  factor? 

68.  By  first  finding  the  roots  of  the  equation  15  x2  —  4  x  —  3  =  0,  find 
all  the  factors  of  the  expression  15  x2  —  4  x  —  3. 

69.  Based  upon  the  note  under  Ex.  3,  and  upon  Exs.  64-68,  write  a 
carefully  worded  rule  for  factoring  quadratic  expressions. 

Apply  the  rule  asked  for  in  Ex.  69  in  finding  all  the  factors  of  the 
following  expressions,  and  verify  their  correctness  : 

70.  5  x2  +  12  X  -  9.  73.    (x  +  1)(2  -  x)  +  9  -  x. 

71.  8.2_io,_3.  ^^    (2,  -  3)2- 6(2, +  1)+ 8. 
„2    3x2      ^ 

'^    ~4"~2~  75.   ax2  +  6x  +  c. 


274  ELEMENTARY  ALGEBBA  [Ch.  XV 

76.  Are  the  expressions  in  Exs.  70-75  equal  to  0  ?  What  justification 
have  we  then  for  writing  them  so  ? 

77.  Write  an  equation  whose  roots  are  3  and  8  (cf.  §  72). 

78.  Write  an  equation  whose  roots  are  —  |  and  12;  7  and  —  1;  f  and 
V- ;  1  +  V3  and  1  -  \/3 ;  i  audi;  2  +  3  i  and  2  -  3  i. 

79.  By  first  finding  the  factors  of  x^  —  2  x  —  10,  prove  that  the  roots 
of  7(x^  —  3  X  —  10)  =  0  are  also  roots  of  a;^  —  3  a;  —  10  =  0,  and  vice  versa. 
Prove  this  also  from  §  95  (2). 

80.  Is  there  any  number  which  is  a  root  ofa;^  —  3a:  —  10  =  0  and  also 
oi  S  x^  +  X  —  10  =  O'j  i.e.,  have  these  equations  a  root  in  common  ? 

Suggestion.  Solve  either  of  these  equations  and  substitute  its  roots  in  the 
other  equation.    Also  solve  by  means  of  §  76. 

81.  Find  the  common  roots,  if  any,  of  2  x^  —  S3  x^  —  5  x  -\-  Q  =  0  and 

6a:8  +  7x2  +  4a;  +  l  =  0. 

82.  Find  all  the  roots  of  the  equations  in  Ex.  81. 

165.   Solution  of   quadratic  equations  by  factoring.      In  §  72   it 

was  shown  how  factoring  may  be  employed  to  solve  algebraic 
equations;  it  will  now  be  shown  that  any  quadratic  equation 
whatever  may  be  solved  by  this  method. 

Ex.  1.     Solve  the  equation  a:^  +  6  x  +  8  =  0. 

Solution.     The  expression  x^  +  6  a:  +  8 

=  x2  +  6  a;  +  (1)2  _  (1)2  +  8  [cf .  §§  70  and  164 

=  a:2+6a;  +  9-9  +  8 

=  (a:  +  3)2  -  1 

=  {(a:  +  3)  +  1}  .  {(.r  +  3)  -  1} 

=  (a;  +  4)  (a:  +  2)  ; 

hence  the  given  equation  is  equivalent  to 

(x  +  4)  (a:  +  2)  =  0, 

which,  in  turn,  is  equivalent  to  the  two  equations 

a:  +  4  =  0  and  a;  +  2  =  0, 

whose  roots  are  —  4  and  —  2,  respectively ;  therefore,  the  roots  of  the 
given  equation  are   —  4  and  —  2. 

Note.  Observe  that  the  plan  of  the  above  solution  is  first  to  transform  the 
expression  a;2  +  6a;  +  8  into  the  difference  of  two  squares,  one  of  which  shall  con- 
tain all  the  terms  involving  x,  and  then  to  factor  the  resulting  expression  by  the 
formula  A^-B^={A  —  B)  (A  +  B). 


164-165]  QUADRATIC  EQUATIONS  276 

Ex.  2.   Solve  the  equation  a;^  —  3  a;  +  1  =  0. 
^  Solution,    x^  - '6 x  +  1  =  x^  -  Sx  +  1-]^ -I^Y+ 1 

=(^-|-^)-(^-|-f) 

hence  the  roots  of  the  given  equation  are  the  same  as  the  roots  of 

i.e.,  they  are  — — and   '—^ — ^• 

Ex.  3.    Solve  the  equation  ax^  +  bx  -{-  c  =  0. 

Solution.     The  expression  aa:^  +  &x  +  c,  whatever  the  values  of  a,  b, 
and  c,  may  be  factored  as  follows  : 


ax"^  +  bx  +  c  =  alx^  +  -  X  +  -\ 


a-l  X  + 


—  4:ac 

J_  _  VW^Jac}      („   .     6     .    Vb^  -  4  ac 
2a  2a 


M-f« 


a<  X  -^ y  '  i  X  -\ — >• 

i  2a  \      I  2a  ) 


hence  the  roots  of  the  given  equation  are  the  same  as  those  of 


,   ft-V62_ 

■4ac> 

'             2a 

-b  +  Vb^- 

-4rtc 

\  2a  /\  '2  a  J 


z>.,  they  are        -^-rv.   -:.„.   ^^^    _  ft  _  V//^  _  4  «c. 
'       "^  2  a  2« 

Since  every  quadratic  equation  is  reducible  to  the  standard  form 
ax^  +  bx-\-c=:0,  therefore  the  solution  of  Ex.  3  shows  not  only 
how  to  factor  any  expression  of  the  form  ax^  +  bx-j-  c,  but  also  that 
every  quadratic  equation  has  two  roots,  and  only  two ;  compare  also 
§  164,  Ex.  40. 


276  ELEMENTARY  ALGEBRA  [Ch.  XV 

EXERCISES 

4.  By  first  finding  the  factors  of  the  expression  x^  —  9  x  -\-  li,  solve 
the  equation  x^  -  9  x  -{•  H  =  0. 

5.  By  first  finding  the  factors  of  15  a:"  —  4  x  —  3,  find  the  roots  of  the 
equation  15  a;^  —  4  a;  —  3  =0. 

6.  Factor  3y'^-2y-20,  and  thus  solve  the  equation  Sy^-2y-20  =  0. 

Factor  the  following  expressions,  both  by  the  method  of  §  164  and 
also  by  that  of  §  165 ;  also  point  out  which  method  is  simpler,  and  why : 

7.  8  /2  _  10  ^  -  3.  10.   5  m2  +  6  m  +  2. 

8.  (a;  -  1)  (2  -  a;)  +  9  -  x.  11.   x^  +  (m  +  n)x  +  mn. 

9.  3  y2  +  4  2/  _  1.  12.   x^+px-\-  q. 

166.   Solution   of   quadratic   equations   by  means  of   a   formula. 

Since  every  quadratic  equation  in  one  unknown  number  may  be 
reduced  to  an  equivalent  equation  of  the  form  ax^  +  te  +  c  =  0 

(§  163),  and  since  the  roots  of  this  equation  are  — — ^— — — — —^ 

whatever  tJie  numbers  represented  by  a,  b,  and  c  (§  165,  Ex.  3,  and 
§  164,  Ex.  3),  therefore  the  roots  of  any  particular  quadratic  equa- 
tion may  be  found  by  merely  substituting  for  a,  b,  and  c,  in  the 
expressions  for  the  roots  of  the  above  general  equation,  those 
values  which  these  coefficients  have  in  the  particular  equation 
under  consideration. 

E.g.,  since  the  roots  of  ax^  +  bx-{-c  =  0  are  —  ^^vft^^  — 4qc^  therefore  the 

2a 
roots  of  3  x2  + 10  K  —  8  =  0   (in  which  a  =  3,  6  =  10,  and  c  =  —  8)  are 


_10j,VlO._4.3.(-8)^    .^^^   -lOj.14,    .^^^  2   ^^^  __^ 
2*3  6  3 

So,  too,  the  roots  of  6  ?/2  + 19  y  —  7  =  0  are 


19J:Vl92-4.(^(-7)     ,         1    ^^^  _7. 
2-6  3  2 


And    the    roots    of    a;2_3a;  +  5  =  0    are        (     '^)^^{     ^)^     4.1-5^ 

2 
Note.  While  the  student  should,  of  course,  be  able  to  solve  quadratic  equa- 
tions without  the  use  of  the  formula  (by  the  method  of  §  164,  or  of  §  165),  he 
is  advised  to  commit  this  formula  carefully  to  memory,  and  henceforth  to  employ  it 
freely  as  in  the  illustrative  examples  above ;  he  will  find  this  well  worth  his  while, 
because  roots  of  quadratic  equations  are  so  very  frequently  required  in  mathe- 
matical investigations. 


165-167]  QUADRATIC  EQUATIONS  277 

EXERCISES 

1.  Write  down  the  formula  for  the  roots  of  ax^ -{-bx  +  c  =  0.  How 
many  values  has  this  expression  ?  Write  two  expressions  which  are 
together  equivalent  to  this  formula. 

2.  Do  these  two  expressions  represent  the  roots  of  ax^  +  hx  +  c  =  0 
for  all  values  of  the  coefficients  a,  b,  and  c,  or  only  for  particular  values  of 
these  letters  ? 

By  means  of  the  above  formula,  write  down  the  roots  of  each  of  the 
following  equations,  verify  their  correctness  in  each  case,  and  point  out 
which  are  real,  which  imaginary,  which  rational,  and  which  irrational : 

3.  a:2-5a:  +  6  =  0. 

4.  3w2_4^_10  =  0. 


6.    (3y  +  l)(2-r)  ^  y(3 - 

-V). 

7.   mx^  +  nx  +  jo  =  0. 

8     -t2——t  —  ~' 

a         n        2n 

9.   If  the  numbers  represented  by  p  and  q  are  such  that  p^>4:q,  are 

the  roots  oi  x^  +  px  +  q  =  0  real  or  imaginary?   What  are  they  if  jt?2<  4  q'i 

10.   W^hat  are   the  roots   of    36  m^x^  +  36  m^nx  -  n^  =  m^(l  -9n^)l 

Show  that  each  of  these  roots  is  real  whatever  integers  or  fractions 

(positive  or  negative)  may  be  represented  by  m  and  n. 

167.  Character  of  the  roots.    It  has  already  been  shown  (§  165) 
that  the  roots  of  the  equation  ax^  -{- bx -\- c  =  0  are 


-6  +  V62-4ac        -,    _6-V62-4ac 

—-r and ; 

2a  2a  ' 

hence,  if  a,  b,  and  c  represent  real  and  rational  numbers,  these 
roots  can  be  imaginary  or  irrational  only  if  V&^  —  4  ac  is  imagi- 
nary or  irrational.  E.g.,  both  roots  are  imaginary  if  6^  —  4  ac  is 
negative. 

The  conditions  for  discriminating   the  character  of  the  roots 
may  be  summarized  thus : 

if  &^  —  4  ac  >  0,  the  roots  are  real,  and  unequal, 
if  &^  —  4  ac  =  0,  the  roots  are  real,  and  equal, 
if  6^  — 4ac<0,  both  roots  are  imaginary, 
and  the  roots  are  rational  only  when  ft-  —  4  ac  is  an  exact 


*  The  expression  b^  —  ^ao  is,  for  this  reason,  usually  called  the  discriminant 
of  the  quadratic  equation. 


278  ELEMENTARY  ALGEBRA  [Ch.  XV 

The  character  of  the  roots  of  any  particular  quadratic  equation 
may,  therefore,  be  determined  by  merely. finding  the  value  of  the 
expression  &^  —  4  ac  for  that  equation. 

E.g.,  the  roots  of  3  x^  —  5a;  — 1  =  0  are  real,  irrational,  andunequal,  because 
here  h^  —  4tac  =  '61  (since  a  =  3,  6  =  —  5,  and  c  =  —  1),  and  y/'dl  is  real  and  irra- 
tional ; 

The  roots  of  3a;2  — 5ccj— 2  =  0  are  real,  rational,  and  unequal,  because  in  this 
equation  y/b^  —  4  ac  =  \/4y  =  ±  7,  i.e.,  it  is  rational ; 

The  roots  of  2a;2  +  5x  —  8  =  4x  — 11  are  imaginary,  because  in  this  equation 
V62  — 4ac  =  V— 23; 

And  the  roots  of  4  x^  — 12  k  +  9  =  0  are  real,  rational,  and  equal,  because  in 
this  equation  h^  —  ^ac  =  0. 

EXERCISES 

1.  If  62  =  4  ac,  what  is  the  value  of  &2  _  4  ac  ?    of  y/h'^  -4:ac  ?  of 


-bjW¥Z±^9    of    -b-Vb-'-^ac,      How,  then,  do  the  two  roots 

2a  2a 

of  ax^  +  6a:  +  c  =  0  compare  when  6^  =  4  ac  ? 

2.  State  verbally  the  condition  that  must  hold  among  the  coefficients 
of  a  quadratic  equation  in  order  that  the  roots  of  that  equation  shall  be 
equal,  —  instead  of  "a"  say  "the  coefficient  of  the  second  power  of  the 
unknown  number,"  etc. 

3.  For  what  value  of  h  will  the  roots  of  3  a;^  —  10  a;  -f  2  ^  =  0  be 
equal  ? 

Suggestion.    The  roots  are  equal  if  (—  10)2  =  4  .  3  •  2  A;.    Why  ? 

4.  Find  the  value  of  m  for  which  mx^  —  6  a;  +  3  =  0  has  equal  roots. 

5.  Find  the  values  of  k  for  which  the  roots  of  3  x^  —  4  ^a;  +  2  =  0  are 
equal. 

6.  For  what  values  of  a  are  the  roots  of  ax^  —  5  aa:  +  11  =  a  equal? 

7.  For  what  values  of  m  are  the  roots  of  a:^  —  3  a;  —  m{x  +  2  a;^  4-  4) 
=  5a;2  +  3  equal? 

Without  first  solving,  tell  whether  the  roots  of  the  following  equations 
are  real,  imaginary,  rational,  equal,  etc.,  and  explain  your  answers  : 

8.  a:2-5a:  +  6  =  0.  11.   ^t^  +  lit  +  11  =  0. 

9.  a:2-6a:  +  9  =  0.  12.    3:^2  +  2  _  1  ^  £-j, 

7  3         6 

10.   3  f2  _  11  f  _  17  =  0.  13.   7  w2  +  4  w  +  1  =  0. 

14.   Are  the  roots  of  the  equation  in  Ex.  13  related  in  any  way  (cf. 
Ex.  9,  §149)? 


167]  qUADRATIC  EQUATIONS  279 

15.  Show  that  if  either  root  of  a  quadratic  equation  is  imaginary, 
then  the  other  root  is  also  imaginary,  and  that  each  is  the  conjugate  of 
the  other. 

16.  For  what  values  of  k  are  the  roots  of  36  x^  —  24  /:a:  +  15  ^  =  —  4 
imaginary  ? 

Solution.    The  roots  of  this  equation  (§  166)  are 


24  fe  +  V(-24  A:)-^-4  •  36  (15  k+^)  ^^^  2ik-  \/(-24  A:)2-4  •  36  (15  k+i)^ 
2-36  2  •  36 


.  2k+V4:k'^  —  15k-4:       ,  2k~V4ck^-15k-4: . 

6  6  ' 

and  these  roots  are  imaginary  for  those  values  of  k  for  which  the  expression 
under  the  radical,  viz.,  4  k'^  —  15k  —  4,  is  negative,  and  for  those  values  only. 

Now  4  A;2  — 15  ^  —  4j  which  equals  (4  A;  + 1)  (k  —  4)  (§  165) ,  is  negative  for  those 
values  of  k  for  which  one  of  these  factors  is  positive  and  the  other  negative,  and 
for  no  others  ;  hence  the  roots  of  the  given  equation  are  imaginary  when  k  lies 
between  —  ^  and  4. 

17.  From  the  solution  of  Ex.  16  point  out  those  values  of  k  for  which 
the  roots  of  the  given  equation  are  real,  and  explain  your  answer. 

18.  If  k  =—  \,  are  the  roots  of  the  equation  in  Ex.  16  real  or  imagi- 
nary ?     How  do  they  compare  in  value  when  k  =  —  ^'i   when  A:  =  4 ?   " 

19.  "Without  actually  solving  the  equation,  find  the  values  of  m  for 
which  the  roots  of  4  m^x"^  +  12  m^x  +  10  —  m  =  0  are  equal. 

20.  Without  actually  solving  the  equation,  find  the  values  of  m  for 
which  the  roots  in  Ex.  19  are  real,  and  those  for  which  these  roots  are 
imaginary. 

21.  Find  the  sum  of  the  two  roots  of  ax^  -\-  hx  -{■  c  =  0 ',  also  the  sum 
of  the  roots  oi  x^  ■}■  px  +  q  =  0. 

22.  By  means  of  the  results  of  Ex.  21,  and  without  first  solving  the 
equation,  determine  the  sum  of  the  roots  of  x^  +  Sx  —  2  =  0;  also  the 
sum  of  the  roots  of  4  x^  —  6  a;  =  3.  Verify  your  answers  by  actually 
adding  the  roots. 

23.  Find  the  product  of  the  roots  oi  x^  +  px  +  q  =  0',  also  the  product 
of  the  roots  of  ax^  -\-  bx  +  c  =  0. 

24.  By  means  of  the  results  of  Ex.  23,  determine  the  product  of  the 
roots  of  a;2  -  10  X  4-  16  =  0 ;  also  of  4  a;2  _  30  a;  +  25  =  0. 

25.  State  verbally  the  relation  between  the  sum  of  the  roots  of  a 
quadratic  equation  and  the  coefficients  of  that  equation  ;  also  make 
a  similar  statement  concerning  the  product  of  the  roots,  —  compare  Exs. 
21  and  23. 


280  ELEMENTARY  ALGEBRA  [Ch.  XV 

168.   Sum  and  product  of  the  roots.     If  r  and  r'  be  employed  to 
represent  the  roots  of  the  equation  ax^  +  6a;  4-  c  =  0,  i.e.,  if 


r  = !— —  and  r  = , 

2  a  2a  ' 

then  by  adding,  and  by  multiplying,  it  follows  that 

"cf.  Exs.  21  and 
23,  §  167 


b  c 

r  -\-7''  = and  r  -  r'  =  - 


a  a 

The  student  should  perform  these  operations  in  detail,  and  should  also  express 
the  results  in  verbal  language.    Compare  Ex.  25,  §  167. 

Note.  Rationalizing  the  numerators  in  the  above  expressions  for  the  roots  of 
az^+bz  +  c  =  0,  shows  that 


-&  +  v/62_ 

-4ac 

-2c 

2a 

6+V62- 
-2c 

4:ac 

-b-Vb^- 

-4  ac 

2a 

6-V62- 

4ttc 

and 

Since  r-r'  =-,  therefore  if  c  is  very  small  as  compared  with  a,  i.e.,  if  -  is  very 
a  a 

small,  then  at  least  one  of  the  roots  (r  or  r')  must  be  very  small ;  to  see  which  one 
this  is,  and  also  to  see  how  large  the  other  root  is,  it  is  only  necessary  to  examine 
the  above  expressions  for  r  and  r'. 

Thus  as  c  =  0,*  4  ac  =^  0,  and  b^  —  4tac^  b^,  i.e.,  VP^^^^Toc  =  b,  and  the  first 

expression  for  r  shows  that  ?•  =  0,  —  since  —  =  0. 

2  a 
Similarly  it  may  be  shown,  from  the  first  expression  for  r',  that  when  c  =  0, 

then  r'  = ,  — observe  that  the  second  expression  for  r'  becomes  indeterminate 

a  r. 

when  c  =  0,  i.e.,  it  becomes  — 

What  has  just  been  shown  is  usually  expressed  by  saying  "if  the  absolute  term 
of  a  quadratic  equation  is  zero,  then  one  root  of  that  equation  is  also  zero*' 
(cf.  Ex.  20,  §  164). 

Again,  if  a  =  0,  then  the  above  expressions  show  that  r'  becomes  —  oo  (cf .  note 
to  Ex.  15,  §  55),  and  that  r  becomes  — -,  —  the  first  expression  for  r  becomes  -, 

c  ^ 

which  is  indeterminate,  but  the  second  shows  its  value  to  be  —  -• 

b 

What  has  just  been  shown  may  be  expressed  by  saying  "  a  =  0  is  the  condition 

that  one  root  of  az^  +  bz  +  c  =  Ois  infinitely  large.'* 


EXERCISES 

1.  Without  solving  the  equations,  write  down  the  sum  and  also  the 
product  of  the  roots  of  each  of  the  equations  in  Exs.  6-11  of  §  164,  and 
explain  your  answer  in  each  case. 

*  The  symbol  =  is  here  used  to  mean  "  approaches  indefinitely  near  to." 


168]  QUADRATIC  EQUATIONS  281 

2.  Give  the  sum  and  also  the  product  of  the  roots  of  each  equation 
in  Exs.  22-27  of  §  164,  and  verify  your  work. 

3.  If  one  root  of  the  equation  x^  +  5  a;  —  24  =  0  is  known  to  be  3, 
how  may  the  other  root  be  found  from  the  absolute  term?  from  the 
coefficient  of  the  first  power  of  x  ?     Do  the  results  agree  ? 

4.  If  one  root  of  any  given  quadratic  equation  whatever  be  known, 
how  may  the  other  root  be  most  easily  found? 

5.  What  is  the  sum  of  the  roots  of  3  nfiz^  +  (8  m  -  l)a:  +  5  =  0  ?  For 
what  value  of  m  is  this  sum  3  ? 

6.  For  whatr  values  of  A;  will  one  of  the  roots  of  2  (k  ■]- lyx^ — 
^(2k-i-  I)  (k  +  l)x+9k  =  0he  tlie  reciprocal  of  the  other  ? 

Suggestion.    Equate  one  of  the  roots  to  the  reciprocal  of  the  other,  and  solve. 

7.  For  what  value  of  k  will  one  root  of  the  equation  in  Ex.  6  be  zero  ? 
With  this  value  of  k,  what  will  be  the  value  of  the  other  root? 

8.  For  what  value  of  k  will  one  root  of  the  equation  in  Ex.  6  be 
infinite  (cf .  note,  §  168)  ? 

9.  For  what  values  of  n  will  one  of  the  roots  of  (n  —  3)y^  —  (2  n  +  1)  y 
=  2  —  5  n  be  double  the  other  ? 

10.  Prove  that  one  of  the  roots  of  ax"^  -\-  bx  +  c  =  0,  whatever  the 
values  of  a,  b,  and  c,  will  be  double  the  other  if  2b'^  =  9  ac. 

11.  If  r  and  r'  are  the  roots  of  ax^  -\-  bx  -\-  c  =  0,  find  the  value  of 

-  +  —  expressed  in  terms  of  a,  b,  and  c. 
r     r' 

12.  It  has  already  been  shown  that  if  r  and  r'  are  roots  of  the  equation 
ax^  +  bx  -\-  c  =  0,  then  ax^  +  bx  -\-  c  =  a(x  -  r)  (x  -  r')  ;  from  this  fact 
prove  that  if  r"  is  not  equal  to  r  or  to  r',  then  r"  can  not  be  a  root  of 
ax^-{-bx  +  c  =  0  (cf.  Ex.  40,  §  164).  

13.  Show  that  the  roots  of  ax^ +  2bx -\- c  =  0  are  -b+^b^-ac  and 


b-Vh^ 


a 


How  do  these  expressions  compare  with  the  expressions 


a 
for  r  and  /  above  ? 

14.  Apply  the  formulas  of  Ex.  13  to  write  down  the  roots  of  3  a;^  —  8  a; 
—  3  =  0;  also  of  2  x^  +  10  a;  =  7.  Compare  these  results  with  those 
obtained  by  the  formulas  of  §  166 ;  which  of  these  formulas  gives  the 
simpler  result  when  the  coefficient  of  the  first  power  of  the  unknown 
number  is  even? 

15.  Show  that  when  a  and  c  represent  numbers  having  like  signs,  the 
roots  of  ax^  +  bx  +  c  =  0  may  be  real,  or  may  be  imaginary,  depending 
upon  the  relative  values  of  a,  6,  and  c ;  but  that  these  roots  are  necessarily 
real  when  a  and  c  represent  numbers  having  unlike  signs. 


282  ELEMENTARY  ALGEBRA  [Ch.  XV 

16.  What  relation  exists  between  the  roots  of  ax^  +  hx  -{-  c  =  0  when 
a  =  c'i    when  a  =  —  c'i 

17.  If  r  and  r'  represent  the  roots  of  ax^  +  6a;  +  c  =  0,  form  an  equation 
whose  roots  are  —  r  and  —  r'. 

Solution.    The  equation  whose  roots  are  —  r  and  —  r'  is  (§  72} 
{x-\-r){x  +  r')=Q,  i.e.,  a:2+ (r  +  r')a;  +  rr' =  0; 

6  c 

but  r-\-r'  = and  rr'  =  -  (§  168) ,  hence  the  required  equation  is 

a;2  — -a;+-  =  0,  i.e.,  ax^—hx-\-c  =  0. 
a        a  X 

18.  Find  r^  +  r'2  from  ax"^  -{-hx  +  c  =  0.  Also  find  the  sum  of  the 
reciprocals  of  the  roots  ofx^  —  5a:  +  2  =  0  (cf.  Ex.  11). 

19.  If  r  and  r'  are  the  roots  of  ax'^  -{■  hx  -\-  c  =  0,  form  the  equation 

whose  roots  are  r^  and  r'^ ;  also  one  wliose  roots  are  -  and  — • 

r  r 

20.  What  do  the  roots  of  nx^  +  hx-[-c  =  0  become  when  c  =  0? 
when  c  =  0  and  6  =  0?  when  a  =  0  ?  when  a  =  0  and  6  =  0?  when  6  =  0? 
Compare  the  note  on  p.  280. 

169.  Fractional  equations  which  lead  to  quadratics.  The  general 
principles  underlying  the  solution  of  fractional  equations  are  dis- 
cussed in  §  99;  manifestly  those  principles  apply  whatever  the 
degree  of  the  integral  equation  to  which  the  fractional  equation 
leads.     The  following  solutions  may  illustrate  the  procedure. 

Ex.  1.     Solve  the  equation  ^  "^     +  1  =  3  x. 

X  +  2 

Solution.     On  clearing  the  given  equation  of  fractions,  it  becomes 

a:  +  5  +  a;  +  2  =  3a;2  +  6a;, 

which  reduces  to        3  a;^  -f  4  x  ~  7  =  0, 

whence  4  j:  Vl6  +  84  ^^ 

6  ^^ 

-4±10 


6 
1  or   - 


and  since  neither  x  =  \  nor  a;  =  —  |  reduces  to  zero  the  multiplier  which 
was  used  to  clear  of  fractions,  therefore  they  are  the  roots  of  the  given 
equation  (cf.  §  99). 


168-170]  QUADRATIC  EQUATIONS  283 

Ex.  2.   Solve  the  equation  -^—  +  *  ^  +  '^  -    ^  ^^ 


1  -  X       X  +  1       x2-l 
Solution.     On  clearing  the  given  equation  of  fractions,  it  becomes 
-x^-x-^x-S  +  ix^  +  Sx  =  2x^, 
which  reduces  to  a;^  —  2  x  —  3  =  0, 


whence  ^  ^  2  ^  V4TT2  ^  2_^^ 

2  2 

I.e.,  x='d  or  —  1 ; 

but  although  both  3  and  —  1  are  roots  of  the  integral  equation,  yet  3 
alone  is  a  root  of  the  given  fractional  equation.  Observe  that  a:  =  —  1 
reduces  the  multiplier  a:^  —  1  to  zero ;  compare  also  §  99. 

EXERCISES 
Solve  the  following  fractional  equations,  being  careful  to  exclude  all 
extraneous  roots : 

3.   15x  +  ?=ll.  -    2x-2_x-l 


X  .5a:+5ar  +  l 


X  X  x  +  2x  —  2 


-C-^) 


^  ^         ^  8.   ^+(x-2)-i 


2(a;2-l)      4(a:  +  l)      8  x -1 

9.  -1^+ 42_ ^g_^     6 


X  +  5      (x  +  5)  (a:  -  2)  a:  -  2 

10  ^Q    I  ^Q         I  7=   ^^  12    2a  +  a;      a-  2  a:  ^  8 

■   a;  +  3     a;2+4:a;  +  3  a;  +  l*  '   2  a  -  a;      a  +  2  a;     3* 

11  2a;  +  l      5^a:-8  ^3       Ix      ^  ^a(a:  +  2  6) 
l-2a:     7         2     *  '  a  -  a;  a  +  6 

3^4        2      _      5a:     ^  a:  +  29 g  • 


15. 


a; -5     3  a; +  2      (3  a;  +  2)  (a:  -  5) 

X X 

X  —  1      a;  +  1 


170.  Irrational  equations.  Equations  which  contain  indicated 
roots  of  the  unknown  numbers  are  usually  called  irrational  equa- 
tions ;  they  are  also  sometimes  spoken  of  as  radical  equations. 

E.g.,  V^-o  =  0,  VF+l  +  a;  =  8,  ^^  +  1  =  0,  and  3  +  ^=\/^2Zri  are 
irrational  equations,  but  x  —  -\/3  =  5  k  is  a  rational  equation. 

The  solution  of  irrational  equations  may  be  illustrated  by  the 
following  examples : 


284  ELEMENTARY  ALGEBRA  [Ch.  XV 

Ex.  1.   Solve  the  equation  Vx  —  5  =  0. 

Solution.    The  given  equation  is  (§  95)  equivalent  to 
Vx  =  5, 
whence,  squaring,  x  =  25. 

On  substituting  25  for  x,  the  given  equation  is  satisfied,  provided  that 
Vx  is  understood  to  mean  the  positive  value  of  the  square  root ;  and  in 
that  case  25  is,  therefore,  a  root  of  the  given  equation. 

Ex.  2.   Solve  the  equation  y/x  +  1  +  a;  =  11. 

Solution.     The  given  equation  is  (§  95)  equivalent  to 

Vx  +  1  =  11  -  x, 

whence,  squaring,  a:  +  1  =  121  —  22  a:  +  a:^, 

t.e.,  a;2  -  23  X  +  120  =  0, 

,                                                            23  ±  V232  -  480 
whence  x  =  — — — > 

i.e.,  X  =  15  or  8, 

and,  on  substitution,  it  is  found  that  15  satisfies  the  given  equation  if 
Vx  +  1  means  the  negative  value  of  this  root,  while  8  satisfies  it  if  the 
positive  value  of  this  root  is  intended. 

Ex.  3.   Solve  the  equation  ^^l£  +  1  =  0. 

Vx 
Solution.     The  given  equation  is  equivalent  to 

6  —  X  =  —  Vx, 
whence,  squaring,  36  —  12  x  +  x^  =  x, 

and  therefore  x  =  9  or  4  ; 

of  which  9  is  a  root  of  the  given  equation  if  the  positive  value  of  the 
square  root  is  meant,  otherwise  4  is  a  root. 

The  above  procedure  may  be  formulated  thus :  (1)  isolate  the 
radical,  or  one  of  the  radicals,  if  there  are  two  or  more, 
(2)  hy  involution  rationalize  the  given  equation,  (3)  solve 
this  rational  equation,  and  (4)  test  the  results  hy  sub- 
stituting them  in  the  given  equation. 

Note  1.  Observe  that  a  quadratic  irrational  equation  is  ambiguous  unless  it 
is  stated  which  of  the  two  values  of  the  radical  is  intended. 

E.g.t  the  equation  Vx  — 5  =  0  really  contains  in  itself  two  equations,  viz., 
Vx  —  5  =  0*  and  Vx  —  5  =  0;  and  the  equation  Vx  +  Vs  —  x  =  3  contains  in  itself 

*  Let  V  and  V  indicate  the  positive  and  negative  values,  respectively,  of 
the  roots. 


170J  QUADRATIC  EQUATIONS  285 

four  equations,  viz.,  \/z -h  ^/5  —  x  =  3,  Va:  +  VS  —  x  =  3,  'y/z  +  V5  —  x  =  3, 
and  Vx  +  Vo  —  X  =  3,  Hence,  in  order  to  avoid  ambiguity,  it  is  always  neces- 
sary to  specify  in  connection  with  a  radical  equation  which  root  is  intended. 

Note  2.  It  sliould  also  be  observed  that  if  both  members  of  any  given  equation 
be  raised  to  the  same  positive  integral  power,  then  every  root  of  the  given  equa- 
tion will  be  a  root  of  the  new  equation  thus  formed,  and  the  new  equation  will,  in 
general,  have  one  or  more  additional  roots  which  were  introduced  by  the  involution. 

To  prove  this,  let  the  members  of  the  given  equation  be  represented  by  u 
and  V  respectively  (where  u  and  v  may  be  expressions  containing  the  unknown 
number  x) ;  then  the  given  equation  is  u  =  v,  and  from  this  it  follows  by  squaring 
that  u^=v^,  which  is  equivalent  to  u^  —  v^  =  0,  i.e.,  to  {u  —  v){u  +  v)=0;  but 
every  root  of  the  given  equation  makes  u  =  v,  i.e.,  makes  u  —  v^O,  and  hence 
satisfies  the  equation  {ii  —  v)  (w  +  u)  =  0,  while  the  new  equation  is  also  satisfied 
by  those  additional  values  of  x  which  make  w  +  u  =  0;  hence  the  correctness  of 
the  above  statement. 

Similarly  if  the  members  of  the  given  equation  had  been  raised  to  a  higher 
power  than  the  second. 

Hence  the  roots  of  any  given  irrational  equation  are  to  be  found  among  the 
roots  of  the  equation  resulting  from  rationalizing  the  given  equation,  and  if  none 
of  the  roots  of  the  rational  equation  prove  to  be  roots  of  the  irrational  equation, 
then  that  equation  has  no  root  whatever. 

E.g.,  the  equation  V'Sx  +  i  +  2Vx  +  5  -  Vx  =  0  leads  to  3  x2  -f  4  x  -  64  =  0, 
whose  roots  are  4  and  —  V>  neither  of  which  is  a  root  of  the  given  equation,  hence 
that  equation  has  no  root  whatever. 

EXERCISES 

4.  Show  that  if  the  signs  of  the  radicals  are  left  unrestricted,  then 
the  equation  V8  a;  -j-  4  +  2  Vx  -f  5  —  Vx  =  0  has  two  roots.  What  are 
these  roots? 

Solve  the  following  equations,  and  show  what  restrictions,  if  any,  must 
be  made  on  the  signs  of  the  radicals  in  order  that  your  results  shall  be 
roots  of  the  equations  : 


5.    \/5-a:  =  a;-5,  11.    V4a:-|-1- Va:-f:3: 


6.   x+V^  =  4x-4Vi.  "•    V^T^+V.  +  6=V2x+«  +  ft. 


13.    Va:  +  3-l-V4a;-l-l  =  Vl0x  +  4. 
^^     V3  a:  +  1  +  V3^ 
8.    \/4!/-f  17  +  \/z/  +  l-  4  =  0.  V3F+T  -  >/3x 


^•^  +  ^^  =  ^«-  ^^^    V,37TT+V8^_, 


9.    Vx  -f  1  -f  (a;  4-  l)-2  =  2.  ^^^ 


Vx-  2_  Vi-f-  1 


Vx  +  3      Vx  +  21 


10.    VS  +  x-l-Vx^A.  i6_  J^%6_>'_ 

\X  ^  X  ^  X 


286  ELEMENTARY  ALGEBRA  [Ch.  XV 

Find  all  the  roots  of  the  following  restricted  equations  (cf.  note  2, 
above),  and  verify  your  results : 

17.    V^in  +  V^^^  =  2.  20.   V3^^+V^^-2v'^^=0. 


18.  VxT4:+Vx^^  =  -2.  21.    V3X-5+ v^^-2Va:-l  =  0. 

19.  vTT5+Vx^^  =  2.  22.   Va^^+Vx^-2Vi^=0. 

23.  By  first  rationalizing  the  equation  x  =  VT,  and  then  transposing 
and  factoring,  show  (§  72)  that  this  equation  has  3  solutions;  i.e.,  show 
that  1  has  3  distinct  cube  roots,  viz. :  1,  i(  — 1+  V^)  and  |(  — 1— V  — 3). 

Similarly  it  may  be  shown  that  any  number  whatever  has  3  cube  roots 
(cf.  §  132). 

171.  Problems  which  lead  to  quadratic  equations.  The  directions 
already  given  for  solving  problems  whose  conditions  lead  to 
simple  equations  (§  26)  are  also  applicable  to  problems  which 
lead  to  quadratic  and  still  higher  equations ;  the  three  important 
steps  are : 

(1)  Translate  the  conditions  of  the  problem  into  equations, 

(2)  Solve  these  equations, 

(3)  Test  and  interpret  the  results. 

Special  emphasis  is  to  be  laid  upon  the  testing  and  interpreting  of 
the  results,  because  a  problem  often  contains  restrictions  upon  its 
numbers,  expressed  or  implied,  which  are  not  translated  into  the 
equations,  and  therefore  the  solutions  of  the  equations  may  or  may 
not  be  solutions  of  the  problem  itself,  —  compare  the  illustrative 
problems  which  follow,  and  also  §  100. 

Prob.  1.  A  farmer  purchased  some  sheep  for  $168  ;  later  he  sold  all 
but  4  of  them  for  the  same  sum,  and  found  that  his  profit  on  each  sheep 
sold  was  |1.     How  many  sheep  did  he  purchase  ? 

Solution 

Let  X  =  the  number  of  sheep  purchased. 

1  ftft 
Then  =  the  number  of  dollars  each  sheep  cost, 

X 

1  OQ 

and =  the  number  of  dollars  received  for  each  sheep, 

a?  —  4 

and  hence      i^  -  1^  =  1 ;  fSi^ee  the  profit  is 

X  —  4:       X  L$lon  each  sheep 

therefore  (§  169)  x  =  28  or  -  24. 


170-171]  QUADRATIC  EQUATIONS  287 

The  first  of  these  values,  viz.,  28,  is  found  to  be  a  solution  of  the  prob- 
lem as  well  as  of  the  equation,  but  while  the  second  satisfies  the  equation 
it  can  not  satisfy  the  problem,  because  the  number  of  sheep  purchased  is 
necessarily  a  positive  integer. 

Prob.  2.  At  a  certain  dinner  party  it  is  found  that  6  times  the  num- 
ber of  guests  exceeds  the  square  of  f  their  number  by  8 ;  how  many  guests 
are  there? 

Solution   ' 

Let  X  =  the  number  of  guests. 

Then  the  expressed  condition  of  the  problem  is 

t.e.,  2  a:2  -  27  X  +  36  =  0, 

whence  a:  =  12  or  f . 

Here,  too,  an  implied  condition  of  the  problem  is  that  the  answer  must 
be  a  positive  integer,  and  hence,  although  f  satisfies  the  equation,  it  is 
not  a  solution  of  the  problem. 

Prob.  3.  If  4  times  the  square  root  of  a  certain  number  be  subtracted 
from  that  number,  the  result  will  be  12 ;  what  is  the  number? 

Solution 

Let  X  =  the  required  number. 

Then  the  problem  states  that  a;  —  4\/x  =  12, 

Le.,  '  a;2  -  40  a;  +  144  33  0, 

whence  a:  =  36  or  4. 

If  the  above  square  root  is  understood  to  be  positive,  then  36  is  the 
solution,  but  if  the  negative  root  is  meant,  then  4  is  the  solution. 

Prob.  4.  If  the  number  of  dollars  that  a  certain  man  has,  be  multi- 
plied by  that  number  diminished  by  4,  the  product  will  be  21.     How 

much  money  has  he  ? 

Solution 

Let  X  =  the  number  of  dollars  he  has. 

Then  the  problem  states  that  x{x  —  4)  =  21, 

whence  a;  =  7  or  —  3. 

Each  of  these  numbers  will  satisfy  the  conditions  of  the  problem,  pro- 
vided, in  the  case  of  the  second,  that  a  negative  possession  be  regarded  as 
an  indebtedness;  i.e.,  the  man  may  either  possess  $7,  or  he  may  owe  |3. 


288  ELEMENTARY  ALGEBRA  [Ch.  XV 

Prob.  5.  The  sum  of  the  ages  of  a  father  and  his  son  is  100  years, 
and  one  tenth  of  the  product  of  the  number  of  years  in  their  ages,  minus 
180,  equals  the  number  of  years  in  the  father's  age ;  what  is  the  age  of 

each  ? 

Solution 

Let  X  =  the  number  of  years  in  the  father's  age. 

Then       100  —  x  =  the  number  of  years  in  the  son's  age, 

and  the  condition  of  the  problem  states  that 

a:(100-.)_^3Q^ 
10 

whence  a;  =  60  or  30. 

Although  each  of  these  numbers  is  a  positive  integer,  yet  the  second 
is  not  a  solution  of  the  problem,  since  it  would  make  the  son  older  than 
the  father.     Hence  the  father  is  60,  and  the  son  40  years  old. 

If,  in  the  above  problem,  "two  persons"  be  substituted  for  "a  father 
and  his  son,"  then  both  solutions  are  admissible,  and  their  ages  are 
either  60  and  40,  or  30  and  70  years. 

PROBLEMS 

6.  Find  two  numbers  whose  difference  is  11,  and  whose  sum  multi- 
plied by  the  greater  is  513. 

7.  A  man  purchased  a  flock  of  sheep  for  $75.  If  he  had  paid  the 
same  sum  for  a  flock  containing  3  more  sheep  they  would  have  cost  him 
$1.25  less  per  head.     How  many  did  he  purchase? 

Is  each  solution  of  the  equation  of  this  problem  a  solution  of  the  prob- 
lem itself?     Why? 

8.  A  clothier  having  purchased  some  cloth  for  $30  found  that  if  he 
had  received  3  yards  more  for  the  same  money,  the  cloth  would  have  cost 
him  50  cents  less  per  yard.  How  many  yards  did  he  purchase?  Has 
this  problem  more  than  one  solution? 

9.  Divide  10  into  two  parts  whose  product  is  22|. 

10.  Find  two  numbers  whose  sum  is  10  and  whose  product  is  42.  Are 
there  any  two  real  numbers  which  satisfy  these  requirements? 

11.  Find  two  consecutive  integers  the  sum  of  whose  squares  is  61. 
How  many  solutions  has  the  equation  of  this  problem?  Show  that  each 
of  these  solutions  of  the  equation  is  also  a  solution  of  the  problem  itself. 


171]  QUADRATIC  EQUATIONS  289 

12.  Are  there  two  consecutive  integers  the  sum  of  whose  squares  is 
118?  Are  there  two  numbers  whose  difference  is  1,  and  the  sum  of 
whose  squares  is  118?  What  are  they?  How  does  the  second  of  the 
above  questions  differ  from  the  first  ? 

13.  Find  three  consecutive  integers  whose  sum  is  equal  to  the  product 
of  the  first  two. 

14.  Is  it  possible  to  find  three  consecutive  integers  whose  sum  equals 
the  product  of  the  first  and  last?  How  is  the  impossibility  of  such  a  set 
of  numbers  shown  ? 

15.  If  the  number  of  eggs  which  can  be  bought  for  25  cents  is  equal 
to  twice  the  number  of  cents  which  8  eggs  cost,  what  is  that  number? 
How  many  solutions  has  the  equation  of  this  problem  ?  Is  each  of  these 
a  solution  of  the  problem  itself  ?    Explain. 

16.  A  farmer,  having  taken  some  eggs  to  market,  found  that  their 
price  had  risen  2|  cents  per  dozen,  and  he  also  discovered  that  he  had 
broken  6  eggs.  He  received  $2.88  for  his  eggs,  which  was  exactly  what 
he  would  have  received  if  he  had  broken  none,  and  if  the  price  had  not 
risen.     How  many  eggs  did  he  take  to  the  market? 

Is  each  solution  of  the  equation  of  the  problem  a  solution  of  the  prob- 
lem itseK  ?    Explain. 

17.  Find  two  numbers  whose  sum  is  f,  and  whose  difference  is  equal 
to  their  product.     How  many  solutions  has  this  problem  ? 

18.  The  product  of  three  consecutive  integers  is  divided  by  each  of 
them  in  turn,  and  the  sum  of  the  three  quotients  is  74.  What  are  these 
integers?     How  many  solutions  has  this  problem?     Explain. 

19.  If  the  product  of  two  numbers  is  6,  and  the  sum  of  their  recipro- 
cals is  II,  what  are  the  numbers?  How  many  solutions  has  the  equation 
of  this  problem?    How  many  solutions  has  the  problem  itself?    Explain. 

20.  A  merchant  who  had  purchased  a  quantity  of  flour  for  |96  found 
that  if  he  had  obtained  8  barrels  more  for  the  same  money,  the  price  per 
barrel  would  have  been  $2  less.  How  many  barrels  did  he  buy?  How 
many  solutions  has  this  problem?    Explain. 

21.  Why  is  it  that  the  solutions  of  the  equation  of  a  problem  are  not 
always  solutions  of  the  problem  itself?  Compare  the  last  paragraph  in 
§171. 

22.  The  area  of  a  rectangle  is  55^  sq.  in.,  and  the  sum  of  its  length 
and  breadth  is  15  in.     How  long  is  the  rectangle  ? 


290  ELEMENTARY  ALGEBRA  [Ch.  XV 

23.  Find  the  length  of  a  rectangle  whose  area  is  464  sq.  in.,  and  the 
sum  of  whose  length  and  breadth  is  16  in. 

What  is  the  interpretation  of  the  imaginary  result  in  this  problem 
(cf.  note  1,  §  100)  ?  Does  an  imaginary  result  always  show  that  the  con- 
ditions of  the  problem  are  impossible  of  fulfillment  (cf .  Prob.  10,  above)  ? 

24.  A  boating  club  on  returning  from  a  short  cruise  found  that  its 
expenses  had  been  $90,  and  that  the  number  of  dollars  each  member  had 
to  pay  was  less  by  4|  than  the  number  of  men  in  the  club.  How  many 
men  were  there  in  the  club  ? 

25.  If  in  Prob.  24  the  expense  of  the  cruise  had  been  $145,  the  other 
conditions  remaining  unchanged,  how  many  members  would  the  club 
contain  ? 

What  is  the  significance  of  the  fractional  and  negative  results  in  this 
problem  ?  Do  such  results  always  indicate  that  the  conditions  of  a  prob- 
lem are  impossible  of  fulfillment  ? 

26.  The  cost  of  an  entertainment  was  $20,  and  was  to  have  been 
shared  equally  by  those  present.  Four  of  them,  however,  left  without 
paying,  and  this  made  it  necessary  for  each  of  the  others  to  pay  25  cents 
extra.     How  many  persons  attended  the  entertainment? 

27.  The  number  of  miles  to  a  certain  city  is  such  that  its  square  root, 
plus  its  half,  equals  12.     What  is  the  distance  ? 

Has  this  problen\  more  than  one  solution?    Explain. 

28.  When  a  certain  train  has  traveled  5  hours  it  is  still  60  miles  from 
its  destination ;  if  it  is  also  known  that,  by  traveling  5  miles  faster  per 
hour,  1  hour  could  be  saved  on  the  whole  trip,  what  is  the  entire  distance? 
And  what  is  the  actual  speed  ? 

29.  The  diagonal  and  the  longer  side  of  a  rectangle  are  together  five 
times  the  shorter  side,  and  the  longer  side  exceeds  the  shorter  by  35  yards. 
What  is  the  area  of  the  rectangle  ? 

30.  It  took  a  number  of  men  as  many  days  to  dig  a  trench  as  there  were 
men.  If  there  had  been  6  more  men,  the  work  would  have  been  done  in 
8  days.     How  many  men  were  there  ? 

31.  A  crew  can  row  5J  miles  downstream  and  back  again  In  2  hours 
and  23  minutes ;  if  the  rate  of  the  current  is  3|  miles  an  hour,  find  the 
rate  at  which  the  crew  can  row  in  still  water. 

32.  A  crew  can  row  a  certain  course  upstream  in  8f  minutes,  and  if 
there  were  no  current,  they  could  row  it  in  7  minutes  less  than  it  takes 
them  to  drift  down  the  stream.  How  long  would  it  take  them  to  row 
the  course  downstream  ? 


171-172]  QUADRATIC  EQUATIONS  291 

33.  The  hypotenuse  of  a  right-angled  triangle  is  10  inches,  and  one  of 
the  sides  is  2  inches  longer  than  the  other ;  required  the  length  of  the 
sides. 

34.  From  a  thread  whose  length  is  equal  to  the  perimeter  of  a  square, 
one  yard  is  cut  off,  and  the  remainder  is  equal  to  the  perimeter  of 
another  square  whose  area  is  |  of  that  of  the  first.  What  is  the  length 
of  the  thread  at  first? 

35.  If  one  train  by  going  15  miles  an  hour  faster  than  another,  requires 
12  minutes  less  than  the  other  to  run  36  miles,  what  is  the  speed  of  each 
train  ? 

36.  A  tank  can  be  filled  by  one  of  its  two  feed-pipes  in  2  hours  less 
time  than  by  the  other,  and  by  both  pipes  together  in  IJ  hours.  How 
long  will  it  take  each  pipe  separately  to  fill  the  tank  ? 

37.  A  man  who  owned  a  lot  56  rods  long  and  28  rods  wide  constructed 
a  street  of  uniform  width  along  its  entire  border,  and  thereby  decreased 
the  available  area  of  the  lot  by  2  acres.  What  was  the  width  of  the 
street? 

38.  Of  two  casks,  one  contains  a  certain  number  of  gallons  of  water, 
and  the  other  |  as  many  gallons  of  wine;  6  gallons  are  drawn  from 
each  cask,  and  then  emptied  into  the  other,  after  which  it  is  found  that 
the  percentage  of  wine  is  the  same  in  the  one  cask  as  in  the  other.  How 
many  gallons  of  water  did  the  first  cask  originally  contain? 

39.  A  and  B  together  can  do  a  given  piece  of  work  in  a  certain  time ; 
but  if  they  each  do  one  half  of  this  work  separately,  A  would  have  to 
work  1  day  less,  and  B  2  days  more,  than  when  they  work  together.  In 
how  many  days  can  they  do  the  work  together  ? 

40.  In  going  a  mile,  the  hind  wheel  of  a  carriage  makes  145  revolu- 
tions less  than  the  front  wheel,  but  if  the  circumference  of  the  hind  wheel 
were  16  inches  greater,  it  would  then  make  200  revolutions  less  than 
the  front  wheel.     What  is  the  circumference  of  the  front  wheel  ? 

172.  Equations  above  second  degree,  but  solved  like  quadratics. 

Two  important  classes  of  equations  of  higher  degree  than  the 
second  can  be  solved  like  quadratics ;  they  are  discussed  below, 
(i)  Equations  in  the  quadratic  form.  Equations  which 
contain  only  two  different  powers  of  the  unknown  number,  the 
exponent  of  one  being  twice  that  of  the  other,  may  all  be  reduced 
to  equivalent  equations  of  the  form  ax^""  +  &ic"  +  c  =  0 ;  such  equa- 
tions are  said  to  be  in  the  quadratic  form,  and  may  be  solved  like 
quadratics. 


4 

L 

y=l  or   -  f, 

c2  =  1  or   -  li 

x  =  ±l  or  ± V - 

1; 

292  ELEMENTARY  ALGEBRA  [Ch.  XV 

Eac.  1.     Solve  the  equation  2  x^(x^  +  1)  =  6  -  z^ 

Solution.  The  given  equation  is  equivalent  to  2  x*  -{-  3  x^  —  5  =  0, 
and  on  putting  y  in  place  of  the  lower  power  of  x,  i.e.,  putting  y  =  x^, 
this  equation  becomes 

2y^  +  3y-5  =  0, 

whence  ^  =  llliJ^SE,  ,;§  igg 

i.e., 

and  therefore 

whence 

i.e.,  the  roots  of  the   given   equation  are :    +  1,   —  1,   +  ^  V—  10,  and 
-  ^  V3T0. 

Ex.  2.     Solve  the  equation  x^  +  6  x^  =  3  +  a;^  —  a;3. 

Solution.  The  given  equation  is  equivalent  to  2  x^  +  5  a;  *  —  3  =  0, 
or,  on  putting  y  ior  x^,to  2y^  +  5y  —  3  =  0-, 

whence 

i.e., 

and  therefore 

whence 

Ex.  3.     Solve  the  equation  \^x^-^5x  +  10  =  2  x^  -  10 a:  +  14. 

Solution.  Since  the  rational  part  of  this  equation  is,  so  far  as  the 
terms  containing  x  are  concerned,  simply  a  multiple  of  the  part  under  the 
radical,  therefore  the  equation  may  be  easily  transformed  into  the  quad- 
ratic form ;  thus,  the  given  equation  is  equivalent  to 


y  = 

— 

o± 

V25  + 

24 

4 

y  = 

h 

or 

-3, 

.1  = 

h 

or 

-3, 

X  = 

i 

or 

-27. 

Va:2  -  5  a;  +  10  =  2  (a;2  -  5  x  +  10)  -  6  ; 

and,  on  letting  y  stand  for  Vx^  _  5  x  +  10,  the  given  equation  becomes 

y  =  2y'^-Q, 

whence  y  =  2  ov  —  f , 

i.e.,  Va;2  _  5  a:  +  10  =  2  or  -  f , 

and  therefore         a;^  —  5  x  +  10  =  4  or  |, 

whence  a:  =  2,  3,  ^.A^^Zl  or  5^:2^. 

2  2 


172]  QUADRATIC  EQUATIONS  293 

EXERCISES 

4.  Show  that  rationalizing  the  equation  given  in  Ex.  3,  leads  to  an 
equation  of  the  4th  degree.  Is  this  rational  equation  easily  reduced  to 
the  quadratic  form?  Of  the  methods  of  §§  170  and  172  which  is  prefer- 
able in  such  equations  ? 


14_       y2        i,  +  1  _  7 


Solve  the  foil 

owing  equations 

5. 

X*- 

8a;2 

+  12  =  0. 

6. 

3d:6 

-4 

v^  =  10. 

7. 

x^  + 

1  _ 

x^ 

8. 

.- 

.yl 

=  6. 

y  +  1         y2         12 

[Observe  that  ^-^  is  the  reciprocal 

of  -^.1 
v  +  lj 


v^     - 


y  +  2      2(y2  +  4)  _  51 


15.    JLJUL^ 


9.   a;2-7a:  +  Va;2-7a;-l-18=24.  3/2^.4'       ^^2  5 

10.    (z2  +1)2  +  4  (x2  +  1)  =  45.  16.   a:4  +  4  a;8  -  8  x  +  3  =  0.' 


11.  x2-5a:  +  2Vx2-ox-2  =  10.  17.   3/*  +  2  ?/3  +  53/2  +  4  ^  ^  60. 

12.  a;-t  +  5  a^'s  +4  =  0.  18.    16  a;*  -  8  a:^  -  31  a;2  +  8  a: 


13.    (12_i\V8(12_iU33.       +15  =  0- 

V  w         /  \u         J  19.   a;8  +  2  a:2  -  9  a:  =  18. 

(ii)  Reciprocal  equations.  An  equation  which  remains  un- 
changed when,  for  the  unknown  number,  its  reciprocal  is  sub- 
stituted, and  the  new  equation  is  cleared  of  fractions,  is  called  a 
reciprocal  equation. 

Reciprocal  equations  of  the  fifth  and  lower  degrees  are  readily 
solved  like  quadratics,  as  is  shown  in  the  following  examples : 

Ex.  1.     Solve  the  equation  ax^  +  &a:2  +  6a;  +  a  =  0. 

Solution.  This  equation  is  equivalent  to  a  (a;^  -f-  1)  -f  bx  (x  +  1)  =0, 
i.e.,  to  (^  +  1)  •  {«  (a:2  -  X  +  1)  +  hx}  =  0, 

which  is  equivalent  to  the  two  equations, 

X  +  1  =  0  and  ax^  —  ax  -{•  bx  +  a  =  0, 
from  which  the  values  of  x  are  easily  found. 

*  By  extracting  the  square  root  of  the  first  member,  show  that  this  equation 
may  be  written  in  the  form  (a;2  +  2  a;  —  2)2  =  1,  from  which  the  complete  solution 
readily  follows. 


294  ELEMENTARY  ALGEBRA  [Ch.  XV 

Ex.  2.     Solve  the  equation  ax*  +  bx^  +  cx^  ■}■  bx  -\-  a  =  0. 
Solution.     This  equation  is  equivalent  to  ax^  -\-bx  +  c-\ h  —  =  0, 

i.e.,  to  a(x^  +  -\  +b(x +  -]-{- c  =  0; 

1      f         1\^ 
and,  remembering  that       x^  +  —  =  i  x  -\-  -  )   —  2, 

this  equation  becomes       a(x+-]    +blx+-\  +  c  —  2a=:0. 
Now,  on  putting  y  for  x  -{--,  this  last  equation  becomes 

X 

aif  +  by  +  (c  -  2  a)  =  0, 
whence        y  =  -  ^>±  ^^'-4a(c-2a)  ^  ^^  ^^^^  ^^^  -j^^  ^^  ^^^ . 

then  af  +  -  =  k.,  and  x  H —  =  ^9, 

X  X         " 

i.e.,  a;2  -  ^^a:  +  1  =  0,  and  x'^  -  k.p:  +  1  =  0, 

whence  the  four  values  of  x  are  easily  found  when  a,  6,  and  c  are  known. 

EXERCISES 

3.  Prove  (from  the  definition)  that  if  ax^  +  bx'^  +  cx^  +  tZx^  +  ea;+/=0 

is  a  reciprocal  equation,  then  a  =  f,  b  =  e,  and  c  =  (/,  or  a  =  — /,  J  =  —  e,     1 
and  c  =  —  d.     Also  generalize  this  result.  ' 

4.  Show  from  Ex.  3,  by  grouping  terms  as  in  Ex.  1,  that  a  reciprocal 
equation  of  odd  degree  contains  the  factor  a:  +  1  or  a:  —  1. 

5.  By  comparing  Ex.  3,  show  that  every  reciprocal  equation  of  even 
degree  may  have  its  terms  grouped  as  in  Ex.  2. 

Solve  the  following  equations : 

6.  2x8 +  3^2+ 3a;  +  2  =  0.  8.   ^^  -  3  ?/»  +  4?/2  =  3?/ -  1. 

7.  a:4  +  a;8-4a;2  +  a;+ 1  =0.  9.   3a;H6a;4-2  a;8-2a;2  +  6a:  +  3  =  0. 

173.   Maximum   and  minimum  values  of   quadratic   expressions.     I 

Evidently  such  an  expression  as  3  +  5  ic  —  a?^  will,  in  general,  have  ^ 
different  values  when  different  values  are  assigned  to  x ;  and  it  is 
often  important  to  determine  the  greatest  or  the  least  value  (i.e., 
the  maximum  *  or  the  minimum  value)  that  such  an  expression  may 
have,  for  real  values  of  the  letter  or  letters  involved  in  the 
expression. 

*  While  this  definition  is  somewhat  narrow,  it  serves  present  purposes  best. 


172-173]  QUADRATIC  EQUATIONS  295 

Ex.  1.     Find  the  maximum  value  of  the  expression  3  +  5  x  —  z^,  for 
real  values  of  x. 

Solution.    Let  m  stand  for  the  numerical  value  of  the  given  expression, 
i.e.,  let  3  -{-  5  X  —  x^  =  m. 

Then  x^  -  5  x  +  m  -  3  =  0, 


,                                 5  ±  V25  -  4(7/1  -  8)      5±V87-4/n  ..  ,«« 

whence  x  = —^ ^ 1  = -^ [§166 

From  this  last  expression  it  is  clear  (§  167)  that  x  will  be  real  only  so  long 

as  4  my>S7,i.e.,  so  long  as  m>>^V  5  hence  the  greatest  value  that  the  given  ex- 

54-  v37— -4w 
pression  may  have,  while  x  is  real,  is  ^^.   Moreover,  since  x=  '-^ — , 

therefore,  x  —  |  when  m  =  V  ;   «-e-)  f  is  the  value  of  x  which  gives  the 
above  expression  its  maximum  value. 

Ex.  2.     Find  the  least  positive  value  of  x  +  - ,  for  real  values  of  x. 

1  ^ 

Solution.    Let  x  +-=  m.  [Wherein  m  is  positive 

X 

Then  x^  -  mx  +  1  =  0, 

1                                                          m  ±  Vm^  —  4 
whence  x  =  — =^= — 


In  order  that  x  may  be  real,  m^  —  4  <  0,  i.e.,  m  <  2;  hence  the  least 
positive  value  of  m  is  2 ;  and  the  corresponding  value  of  x  is  1. 

Note.    This  exercise  may  also  be  solved  thus:    for  any  real  value  of  x, 
(x  — 1)2<0,  i.e.,  x2  — 2a;  +  l<0,  whence  cc2 4- 1 ^ 2 x-,  whence  x  +  -<2  — since 

the  problem  requires  that  x  be  positive  (why?)  — i.e.,  2  is  then  the  least  value  of 
x  +  -  ;  and  the  expression  takes  this  value  when  x  =  l. 

X 

Ex.  3.     Find  the  range  of  values  of  the  fraction  '       ",  for 

real  values  of  x. 
Solution.    Let 

Then 


a:2_6a:  +  2_ 

x+1 

;2   _ 

-  (6  +  m)x  +  2  -  m  = 

whence      x  =  ^  +  ^  ±^(^  +  ^0' -  ^("- ^)  =  6  +  m  j:  Vm2+ 16  >»  +  28 

Hence,  in  order  that  x  may  be  real, 

m2  +  16  m  +  28  <  0, 
i.e.,  (w  + 14)  •  (m  +  2)<0, 

and,  therefore,  the  factors  m  +  14  and  m  +  2  must  both  be  positive  or 
both  be  negative  (in  order  that  their  product  shall  be  positive)  ;  hence  m 


8     " 


296  ELEMENTARY  ALGEBRA  [Ch.  XV 

may  have  any  value  whatever  from  -co  to  —  14,  and  from  —  2  to  +  oo, 
but  it  can  not  have  a  value  between  —  14  and  —  2.  In  other  words,  for 
real  values  of  x  the  given  fraction  has  no  value  between  —  14  and  —  2. 

Ex.  4.  A  window  consisting  of  a  rectangle  surmounted  by  a  semi- 
circle, is  to  have  a  perimeter  of  18  ft. ;  what  shall  be  the  dimensions  of 
the  rectangle  in  order  that  the  window  shall  admit  the  maximum  amount 
of  light?     And  what  will  be  the  window's  area? 

Solution.  Let  x  stand  for  the  number  of  feet  in  the  width  of  the  win- 
dow; *  then  -  is  the  radius  of  the  semicircular  part,  and  tt-  is  the  semi- 
circle's length.  And  since  the  entire  perimeter  is  18  ft.,  therefore  the  height 
of  the  rectangular  part  must  be  |  (  18  —  x  —  tt-  j ,  i.e.,  9  —  2!_Jl_  x. 

From  these  dimensions  it  follows  at  once  that  the  area  of  the  window  is 

hence,  if  a  represents  the  area, 

9  a:-  "^^-t^^a  =  a, 
8 

whence  (tt  +  4)a:2  -  72  x  +  8  a  =  0. 

Solving  this  equation  gives 

^^36±V(36)2-8a(,r  +  4) 
7r  +  4 
and  hence,  in  order  that  x  be  real, 

(36)'-^  -  8  a(7r  +  4)<  0,  i.e.,  a>  — ^^^^,  which  is  22.68  (nearly)  ; 

8(7r  +  4) 

hence  the  maximum  area  of  the  window  is  nearly  22.68  sq.  ft. ;  and  the 
width  and  height  are,  therefore,  (nearly)  5.04  ft.  and  2.52  ft.,  respectively. 

EXERCISES 

For  real  values  of  x,  find  the  maximum,  or  the  minimum,  value  of  each 
of  the  following  expressions ;  also  the  corresponding  value  of  x : 
5.   x^-8x+  10.  6.    9  -  2a;2  +  16a:.  7.    12  +  x^  -2ax. 

8.  Find  the  range  of  values  of  ^^  +  ^  ^  -  ^. 

9.  Find  the  dimensions  of  the  largest  rectangular  field  that  can  be 
inclosed  by  160  rods  of  fence.     How  many  acres  does  this  field  contain  ? 

*  The  student  should  draw  a  figure  to  represent  the  window ;  it  will  make  the 
solution  easier  to  understand. 


173-175]  QUADRATIC  EQUATIONS  297 

10.  Solve  Ex.  9  if  a  be  substituted  for  160. 

11.  Divide  20  into  two  parts  such  that  the  sum  of  their  squares  shall 
be  a  minimum. 

12.  A  man  who  can  row  4  miles  per  hour,  and  can  walk  5  miles  per 
hour,  is  in  a  boat  3  miles  from  the  nearest  point  on  a  straight  beach,  and 
wishes  to  reach  in  the  shortest  time  a  place  on  the  shore  5  miles  from 
this  point.     Where  must  he  land? 


II.  QUADRATIC  EQUATIONS  IN  TWO  OR  MORE  UNKNOWN 
NUMBERS 

174.  Introductory  remarks.  The  really  essential  thing  in  solv- 
ing any  system  of  simultaneous  equations,  is  first  to  combine  the 
given  equations  so  as  to  eliminate  all  but  one  of  the  unknown  num- 
bers, and  then  to  solve  the  resulting  equation  containing  that 
unknown  number.  When  each  equation  of  the  given  system  is  of 
the  first  degree,  this  elimination,  as  well  as  the  solution  of  the 
resulting  equation,  is  easily  effected  (§  112)  ;  but  these  operations 
become  much  more  difficult  if  one  or  more  of  the  given  equations 
is  quadratic,  or  of  a  still  higher  degree. 

The  next  few  articles  are  devoted  to  a  study  of  the  procedure  in 
cases  where  the  given  system  consists  of  two  equations  one  or 
both  of  which  are  quadratic. 

175.  One  equation  simple  and  the  other  quadratic.  In  this  case 
elimination  by  substitution  (cf.  §  107)  is  usually  advisable. 

Ex.  1.   Solve  the  following  system  of  simultaneous  equations : 

3:^-2^=3,    1  (1) 


,.l 


X^  +  4:f=ld.}  (2) 

Solution.    From  Eq.  (1),     x  =  i±-^,  (3) 

o 

whence,  by  substituting  this  value  of  x,  Eq.  (2)  becomes 

+  4  2/2  =  13,  (4) 


(H^J 


and,  on  expanding  and  simplifying,  Eq.  (4)  becomes 

lOy^+Sy-27  =  0,  (5) 

whence  (§  164)  y  =  I  or  -  |.  (6) 


298 


ELEMENTARY  ALGEBRA 


[Ch.  XV 


But  Eq.  (3)  —  also  Eq.  (1)  —  shows  that  to  every  value  of  y  corresponds 
one,  and  only  one,  value  of  x ;  and  that  when  ?/  =  f  then  x  =  2,  and  when 
y  =  —  f  then  x  =  —  ^.  It  is,  moreover,  easily  verified  that  each  of  these 
pairs  of  numbers  is  a  solution  of  the  given  system  of  equations. 

Manifestly  the  above  method  is  applicable  whenever  one  equa- 
tion of  the  given  system  is  simple  and  the  other  quadratic. 

EXERCISES 

Solve  the  following  systems  of  equations  and  verify  the  correctness  of 
your  results : 


^     (4x-\-Sy  =  9, 
[2x^  +  5x!j  =  3. 

^     (x^  +  xy- 
[x-y  =  '2. 

.     r(^  +  3)(.y-7)=48, 
*•    [x  +  y=lS. 

.     (  2s  +  St  =  10, 


12  =  0, 


uv  —  V  =  10  u, 

+  2  =  v. 


^     (2x^-\-y^  =  3xy-^U, 
'    [2x-y  =  7. 

(1Q  +  4:v  +  2u^  =  5uv, 
[llv-5u  =  4.. 


9. 


10. 


x^  -\-  2  X  +  y  _  4 
^2  _  5  X  +  3  ~  9 

xy 
2      1 


+H=i+4 


=  7.* 


11.  Write  a  rule  for  solving  a  pair  of  simultaneous  equations  one  of 
which  is  simple  and  the  other  quadratic,  and  which  contain  two  unknown 
numbers.  Could  two  such  equations  containing  three  unknown  numbers 
be  solved?    Compare  §  111  note,  and  explain. 

12.  How  many  solutions  has  each  of  the  above  systems  of  equations 
(Exs.  2-10)  ?  Has  every  such  system  two  solutions,  and  only  two  ?  Why 
(see  also  §  176,  Exs.  1  and  2)  ? 

176.  Principles  involved  in  §  175.  The  success  of  the  method 
of  solution  employed  in  §  175  depends  upon  the  fact  that,  if  X,  Y, 


*  Solve  first  for  -  and  1- 
X  y 


175-177]  QUADRATIC  EQUATIONS  299 

and  Z  represent  any  expressions  whatever  which  contain  either 
X  or  y,  or  both,  then  the  system  of  equations 

I  Y'Z=0,} 
is  equivalent  to  the  two  systems 


r.::;)  -  If:::! 


To  prove  this  equivalence,  it  need  only  be  observed  that  every  solution  of 
either  of  the  last  two  systems  is  evidently  a  solution  of  the  first  system;  and 
every  solution  of  the  first  system  is  found  among  the  solutions  of  the  last  two 
systems,  for  it  must  make  X=0  and  also  either  F=  0  or  Z  =  0.* 

EXERCISES 

1.  By  means  of  the  proof  just  given  show  that  Ex.  1,  §  175,  has  two 
solutions,  and  only  two. 

Suggestion.  The  given  system  of  equations  is  equivalent  to  Eqs.  (1)  and  (5) 
(Why  ?),  and  Eq.  (5)  may  be  written  in  the  form  (2  ?/  —  3)  (5  ?/  +  9)  =  0.  Compare 
also  §  108  (iii)  and  §  111. 

2.  By  means  of  the  suggestion  just  given  show  that  every  system  con- 
sisting of  two  equations,  one  of  which  is  simple  and  the  other  quadratic, 
and  containing  two  unknown  numbers,  has  two  solutions,  and  only  two. 

3.  Show  that  the  solutions  mentioned  in  Ex.  2  may  be  imaginary 
(cf.  Ex.  6,  §  175),  and  also  that  one  or  both  of  these  solutions  may  be 
infinite  (cf.  note,  §168). 

4.  In  the  solution  of  Ex.  1,  §  175,  are  Eqs.  (2)  and  (6)  equivalent  to 
the  given  system  ?  May  then  the  values  of  y  from  Eq.  (6)  be  substituted 
in  Eq.  (2)  to  find  the  corresponding  values  of  x?  In  which  two  equations 
may  they  be  substituted?  Why?  Does  your  "rule"  (Ex.  11,  §  175)  pro- 
vide for  this  ? 

177.  Both  equations  quadratic,  —  one  homogeneoust.  If  both  of 
the  equations  of  a  given  system  are  quadratic,  then  elimination 
by  substitution,  as  in  §  176,  leads  to  an  equation  of  the  4th  degree 

(  W  •  X  =  0   ) 
*  Similarly  it  may  be  shown  that  the  system  J   „    7  _  n'  [  ^^  equivalent  to  the 

four  systems  {    j.^^|,    j^^^j.    j  j.  ^  ^^  j  ,  and  j  ^  ^  J  . 

t  An  equation  is  said  to  be  homogeneous  if  all  of  its  terras  are  of  the  same 
degree  in  the  unknown  numbers  (cf.  §  41). 


300  ELEMENTARY  ALGEBRA  [Ch.  XV 

» 
in  one  of  the  unknown  numbers,*  and  this  equation  can  not,  in 
general,  be  solved  by  the  methods  already  studied. 

If,  however,  one  of  the  given  equations  is  homogeneous,  then  the 
solution  of  the  system  may  always  be  made  to  depend  upon  the 
solution  of  a  quadratic  equation  in  one  unknown  number ;  this  is 
illustrated  below. 

Ex.  1.     Solve  the  following  system  of  equations : 

r6x2^-5xy-6^/2  =  0,|  (1) 

\      2x^-y^+5x  =  9.l  (2) 

Solution.     On  dividing  Eq.  (1)  by  y%  it  becomes 

6(^y+5(.^)-6  =  0,  (3) 

whence  (§  164)  ^  =  |,  or  ^  =  -  |,  (4) 

y     d         y         2 

i.e.,  x  =  ly,orx  =  -^y.  (5) 

On  substituting  the  Jirst  of  these  two  values  of  x,  viz.,  |y,  in  Eq.  (2), 
that  equation  becomes 

2(|y)2-3/2  +  5(fy)=9,  (6) 

i.e.,  y'-30y  +  81  =  0,  (7) 

whence  (§  164)  y  =  27  or  y  =  3,  (8) 

and,  since  x  =  ^y,  the  corresponding  values  of  x  are  18  and  2. 

By  substituting  these  pairs  of  numbers,  viz.,  x=18,  y  =  27,  and  x  =  2, 
y  =  3,  in  the  given  system  of  equations,  it  is  easily  verified  that  each  pair 
is  a  solution  of  that  system. 

Similarly,  if  the  second  of  the  two  values  of  x  in  Eq.  (5),  viz.,  —  f  y,  be 
substituted  in  Eq.  (2),  two  other  solutions  of  the  given  system  of  equa- 
tions will  be  found;  these  are :  x  =  —  ^,  y  =  S,  and  x  =  f ,  y  =  —  f . 

It  is,  moreover,  evident  that  every  such  system  of  equations  may  be 
solved  by  this  method. 

Note  1.  The  success  of  the  method  of  solution  here  employed  is  due  to  the 
fact  that  the  two  systems  of  equations  from  which  the  values  of  x  and  y  were 
finally  found,  are  together  equivalent  to  the  given  system. 

*  For  example,  given  the  system  x^  —  Sx  +  Sy  =  4:  and  Sx^  —  Ifi y^  +  20 y  =  9. 

Solving  the  second  of  these  equations  for  y  gives  y  =  i  (5  ±  \/l2a:^—  11),  and 
on  suhstituting  this  value  of  y,  Eq.  (1)  becomes  a;2  _  3  a-  -j-  5  -j-  Vl2  a;2  — 11  =  4, 
which,  when  rationalized,  is  x*  —  6  a;8  —  a;2  —  6  a;  + 12  =  0. 


177-178]  QUADRATIC   EQUATIONS  301 

This  equivalence  may  be  seen  by  writing  the  given  system  thus : 

and  recalling  that,  by  §  176,  this  system  is  equivalent  to  the  two  systems 

[2a;2  — 2/2  +  5x  =  9,  J  \2x^  —  y^  +  5x  =  9, ) 

from  which  the  above  solutions  were  obtained. 

Moreover,  since  each  of  these  systems  has  two  solutions,  and  only  two  (§  176), 
therefore  the  given  system  has  four  solutions,  and  only  four. 

Note  2.    In  practice  the  above  method  may  be  somewhat  simplified  by  putting 
a  single  letter,  say  v,  in  place  of  the  fraction  -  in  Eq.  (3),  i.e.,  by  putting  x  =  vy 

y 

in  the  homogeneous  equation.    Thus,  on  substituting  vy  for  x  in  Eq.  (1),  it  becomes 

6  «2j/^+  5  ?;|/2  _  6  2/2  =  0, 

and  hence,  dividing  by  ?/2,  6  ?;2  -|-  5  ^  —  6  =  0, 

whence  (§  l&i)  w  =  f  or  w  =— f ; 

and,  since  x  =  vy,  therefore  a;  =  |  y  and  x=~^y.    From  here  on  the  work  is  the 

same  as  that  already  given. 

EXERCISES 

Solve  the  following  systems  of  equations  and  verify  the  correctness  of 
your  results  : 

5x'^  +  ^xij  =  y%  (  2{x^ -}- f)  =  5 xy, 

a;2  +  3x  =  5+ I/.  '   \x^-y^=^75. 

^     fx^^xij-U  =  y-x,  ^     (x^-2xy-Sy^  =  0, 

[2x^-3y^  =  xy.  '  \  y(x -\- y)  =  ^. 

6.  Show  that  every  such  system  of  equations  as  those  above  has  four 
solutions  (real  or  imaginary,  finite  or  infinite),  and  only  four. 

178.   Both  equations  homogeneous   in  the  terms  containing  the 
unknown  numbers.    The  solution  of  a  system  consisting  of  two  quad- 
ratic equations,  each  of  which  is  homogeneous  m  the  terms  which 
contain  the  unknown  numbers,  is  easily  made  to  depend  upon  §  177. 
Ex.  1.     Solve  the  following  system  of  equations : 

I  3^2  +  3x^  +  22/2  =  8,  (1) 

I  a;2  _  X2/  -  4  2/2  =  2.  (2) 

Solution.     On  subtracting  Eq.  (1)  from  4  times  Eq.  (2),  the  result  is 
a;2  _  7  ^y  _  18  ^2  ^  0,  (3) 

and  the  given  system  of  equations  is  equivalent  to  the  system  consisting 
of  Eq.  (3)  together  with  either  Eq.  (1)  or  Eq.  (2)  ;  but  of  this  last  system 
Eq.  (3)  is  homogeneous,  and  hence  the  system  can  be  solved  by  the  method 
of  §  177. 


302  ELEMENTARY  ALGEBRA  [Ch.  XV 


r  x2  -  7 

i.e.,  solve  the  equations    \    „ 


EXERCISES 
2.    By  the  method  of  §  177  complete  the  solution  of  Ex.  1  above, 

xy  -42/2  = 
and  verify  the  correctness  of  your  results       - 

Solve  the  following  systems  of  equati(        and  verify  your  results : 
3     (4:X^-xy-S2f  =  2,  ^    ^  (  y^ -\- 15  =  2  xy, 

[x^+Qxy-y''  =  ~Q,  ^^,  \x^  +  y^  =  21  +  xy. 


(2x^-xy  =  28,  g     I 

ta:2  +  2  2/2=18.  '     [ 


2x^-xy  =  28,  ^     {x'^+6xy  =  3-6y% 

x^-2o  =  2y(y  -\-2x). 


7.  Substitute  vy  for  x  in  each  of  the  equations  of  Ex.  6 ;  then  solve  each 
of  the  resulting  equations  for  y^  in  terms  of  v ;  from  the  first  equation 

3                                                                 25 
you  will  find  y^  =  — — ,  and  from  the  second,  y^  =  — — -;  now 

equate  these  two  values  of  y%  solve  the  resulting  equation  in  v,  and  from 
its  values  find  the  values  of  y,  and  thence  the  corresponding  values  of  x. 

8.  Solve  Exs.  4  and  5  above,  by  the  method  outlined  in  Ex.  7. 

9.  Is  the  method  of  Ex.  7  easier  or  more  difficult  than  that  outlined 
in  Ex.  1  ?    In  what  respect  ? 

10.  Is  the  method  of  Ex.  7  applicable  to  all  such  exercises  as  those 
given  above? 

(  Sx^-  5xy-4:y^  =  3x, 

11.  Solve  the  system  ^ 

•^  [9x^-\-  xy-2y^  =  Qx. 

Suggestion.    Subtract  the  second  of  these  equations  from  twice  the  first,  and 
then  proceed  as  in  Exs.  1  and  2  above. 

12.  By  the  method  of  Ex.  11,  solve  the  following  system  of  equations, 
and  verify  your  results : 

(4:X^  +  Qxy-y^  =  ly, 

\Qx^  -Qxy  -\-2y^  =  2y. 

13.  Show  that  the  method  suggested  in  Ex.  11  may  be  successfully 
applied  to  any  system  of  equations  whatever  of  the  form 

ax^  +  hxy  +  cy^  =  dx, 
a'x^  +  b'xy  +  c'y^  =  d'x. 

14.  Could  the  method  suggested  in  Ex.  7  be  employed  in  such  systems 
of  equations  as  those  given  in  Exs.  11,  12,  and  13  ?     Explain. 


178-179J  QUADRATIC  EQUATIONS        ^^^  .  303 

Solve  the  following  systems  of  equations,  and  verify  your  results : 
c^  —  XT/  —  y^  =  2 
:2  -  3  x?/  +  2  3/2 


r2  -  xy  -  ?/2  =  20,  (u^  +  3uv  +  v^=  61, 

^^'     ^  -  -     -         -  '  I  m2  _  y2  =  31  _  2  My. 


2        3 
17.    i 

I  4_         4y2  _  y2  +  2  xy 

[3        !;-l~2(l-a;)' 

179.  Special  devices.  .  kinds  of  systems  of  equations  speci- 
fied in  §§  175,  177,  and  1  >ccur  frequently,  and,  although  they 
present  themselves  in  a  grt.  u  variety  of  forms,  they  may  always  be 
solved  by  the  methods  there  given. 

It  is  worth  remarking,  however,  that  special  devices  of  elimina- 
tion sometimes  give  simpler  and  more  elegant  solutions,  not  only 
for  the  systems  already  considered,  but  also  for  many  others  which 
need  not  now  be  classified.  Some  of  these  special  devices  are 
illustrated  in  the  following  examples,  where  it  is  also  shown  that 
they  apply  to  some  exercises  in  which  equations  above  the  second 
degree  are  involved. 

Facility  in  the  use  of  these  special  devices  can  be  acquired  only 
by  practice,  but  a  little  study  of  any  particular  problem  will  often 
suggest  a  suitable  method  for  attacking  it. 

x-y  =  6,  (1) 

xy  =  -Q.  (2) 

Solution.     From  Eq.  (1),  x'' -  2  xy  +  y^  =  25,  (3) 

fromEq.  (2),  4  2:3/  = -24,  (4) 

adding  Eq.  (4)  to  Eq.  (3),       x^  +  2  xy  +  y"^  =  1,  (5) 

whence  x  +  y  =  ±1;  (6) 

and  from  Eq.  (1)  and  Eq.  (6),  a;  =  3  or  2. 

The  corresponding  values  of  y  are  ?/  =  —  2  or  —  3. 

Observe  that  this  exercise  belongs  to  the  class  of  §  175,  and  could  have  been 
solved  by  the  method  tlere  given. 

+  3a;2/  =  54,  (1) 

a:?/ +  4  2/2  =  115.  (2) 

Solution.    On  adding  Eqs.  (1)  and  (2),  we  obtain 
a:2  +  4  a:?/  +  4  ?/2  =  169, 
i.e.,  ix  +  2yy=m,  (3) 

whence  x  +  2  y  =  ±  13.  (4) 


Ex.  1.    Solve  the  equations   1  ^  ^  ^o> 


f  x^ 
Ex.  2,    Solve  the  equations    \ 


304  ELEMENTARY  ALGEBRA  [Ch.  XV 

From  the  first  of  the  two  equations  in  (4),  and  either  Eq.  (1)  or  Eq.  (2), 
by  §  175,  it  is  found  that  x  =  S,  y  =  5  and  x  =  3Q,  y  =  —  11^  are  solutions. 
Similarly,  by  using  the  second  equation  in  (4),  it  is  found  that  x'=  —  36, 
y  =  11 J  and  x  =  —  3,  y  =  —  5  are  also  solutions  of  the  given  system  of 
equations. 

Observe  that  this  exercise  belongs  to  the  class  of  §  178,  and  could  have  been 
solved  by  the  method  there  given. 

r  a:2  +  2/2  =  6,  (1) 

Ex.  3.    Solve  the  equations    \  ^ ,  ^      ^ 

^  [xy  =  2(x-\-y)-5.  (2) 

Solution.     On  adding  2  times  Eq.  (2)  to  Eq.  (1),  we  obtain 

x^  +  2xy  +  y^  =  ^x  +  y)-^y  -■    ^        (3) 

i.e.,                       (x  +  3/)2  -  4(a:  +  2/)  +  4  =  0 ;     W^\^  )  ^  ^J            (4) 

whence                                                x  -\-  y  =  2.      ^-  (5) 

Substituting  this  value  oi  x  -{■  y  in  Eq.  (2)  gives 

xy  =  ^-5  =  -l;  (6) 

and  2  times  Eq.  (6)  subtracted  from  Eq.  (1)  gives 

x^-2xy  +  y^  =  S,  '           (7) 

whence                                                 x  -  y  =±2\/2,  (8) 

From  Eq.  (5)  and  Eq.  (8),  it  follows  that  x  =  1  +  a/2,  y  =  1  -  V2,  and 
X  =  1  —  V2,  y  =  1  +  a/2  are  solutions  of  the  given  equations. 

Equations  like  those  in  Ex.  3,  which  are  not  changed  by  inter- 
changing X  and  y,  are  usually  said  to  be  symmetric  with  regard  to 
those  letters. 

If  the  equations  of  a  given  system  are  symmetric,  or  symmetric 
except  for  the  signs  of  one  or  more  terms,  their  solution  is  often 
facilitated  by  substituting  u+v  for  one  of  the  letters  and  u—v  for 
the  other ;  this  method  of  solution  is  illustrated  in  Exs.  4-6  below. 

f  x^  +  w2  =r  6,  (1) 

Ex.  4.    Solve  the  equations    ■!  '  '  .„. 

\xy  =  2(x  +  y)-5.  (2) 

Solution.  On  putting  x  =  u  -{-  v  and  y  =  u  —  v,  the  given  equations 
become,  respectively, 

2  m2  +  2  y2  =  6,  and  m2  _  „2  =  4  ^^  _  5 .  (3) 

therefore,  eliminating  i?2  and  simplifying, 

u2  -  2  w  +  1  =  0, 
whence  -  w  =  1. 


179]  QUADRATIC  EQUATIONS  306 

Substituting  this  value  of  u  iu  either  one  of  Eqs.  (3),  gives 
v=±V2, 
whence  (since  x  =  u  +  v,  and  y  =  u  —  v) 

X  =  1  ±  \^,  and  3/  =  1  T  V2, 
which  agrees  with  the  result  found  in  Ex.  3  above. 

Ex.  5.   Solve  the  equations   -i  ' 

\x-y  =  5.  (2) 

Solution.     On  putting  x  =  u  +  v,  and  y  =  u  —  v^  the  given  equations 
become,  respectively,     ^2  _  „2  ^  _  g,  and  2  .  =  5.  (3) 

From  the  second  of  these,  w  =  |, 

and  substituting  this  in  the  first  gives 

whence  a:  =  3  or  2,  and  y  =  —  2  or  —  3  (cf .  Ex.  1,  above) . 

„      ^     o  1        ,  .  [  x^  +  y^  =  xy  —  h, 

Ex.  6.    Solve  the  equations    \ 

1  X  +  ?/  +  1  =  0. 

Solution.      On  putting  x  =  u  +  v  and  y  =  u  —  v,  the  given  equations 
become,  respectively, 

2  w3  _f.  6  My2  _  ^2  +  y2  4.  5  ^  0,  and  2  u  +  1  =  0. 
From  the  second  of  these  equations, 

u  =-  i, 
and  substituting  this  value  in  the  first  gives 

y  =±  f, 
whence  x  =  1  or  —  2,  and  ?/  =  —  2  or  1. 

x^  +  y^=  17,  (1) 


Ex.  7.    Solve  the  equations    ,  „ 

'     x  +  y  =  3.  (2) 

Solution.     This  example  may  be  solved  like  Exs.  4,  5,  and  6 ;  another 

solution  is  as  follows  : 

On  raising  each  member  of  Eq,  (2)  to  the  4th  power,  we  obtain 

x^ -\-  i  x^y  -\- 6  xhf  +  4  a:2/3  +  y4  ^  81,  (3) 

whence,  by  subtracting  Eq.  (1)  from  Eq.  (3)  and  simplifying, 

xy  (2  x2  +  3  a:?/  +  2  y"^)  =  32  ;  (4) 

from  Eq.  (2),  2  x2  +  3  xy  +  2  y^  =  18  -  xy,  (5) 

whence,  on  substituting  from  Eq.  (5),  Eq.  (4)  becomes 

x^/(18-xy)=32,  (6) 

Le.,  (xyy  -  18  (xy)  +  32  =  0,  (7) 

whence  (§  164)  xy  =  2  or  16.  r  (8) 


,   x8-8=  (X2-/)V,  (1) 

Ex.  9.   Solve  the  equations    ' 


306  ELEMENTARY  ALGEBRA  [Ch.  XV 

By  combining  Eq.  (8)  with  Eq.  (2)  it  is  now  easy  to  show  that 

X  =  1,  2,  or  ^ , 

and  the  corresponding  values  of  y  are 

O      ^    y/ KK 

y  =  2,  1,  and      ^     ,  respectively. 

If  one  of  two  equations  is  exactly  divisible  by  the  other,  mem- 
ber by  member,  their  solution  may  often  be  greatly  simplified,  as 
is  shown  below. 

ra;2-3/2=3,  (1) 

Ex.  8.   Solve  the  equations    \  ^  ^^. 

[    x-y=l.  (2) 

Solution.     On  dividing  Eq.  (1)  by  Eq.  (2),  member  by  member,  we 

obtain  3,  ^  2/  =  3,  (3) 

whence,  from  Eqs.  (2)  and  (3), 

X  =  2,  and  y  =  1. 

r  x8  -  8  = 

[  x  +  y  =  2.  (2) 

Solution.     By  transposing,  Eq.  (2)  becomes 

x-2=-y,  (3) 

and,  dividing  Eq.  (1)  by  Eq.  (3),  member  by  member,  we  obtain 

a;2  +  2  a:  +  4  =  -  a;2  +  ?/2,  (4) 

whence,  from  Eqs.  (2)  and  (4),  by  §  175, 

a;  =  0  or  —  6,  and  ^  =  2  or  8. 

Note.  That  this  method  of  division  must  be  applied  with  some  caution  is, 
however,  evident  from  Ex.  9,  for,  while  it  is  easily  verified  that  the  two  pairs  of 
numbers  there  found  are  solutions  of  the  given  system  of  equations,  that  system 
has  another  solution,  viz.,  x  =  2,  and  y  =  0,  which  the  above  process  has  failed  to 
reveal.  This  last  solution  is  found  by  equating  each  member  of  Eq.  (3)  sepa- 
rately to  zero.* 

*  The  general  theory  for  such  cases  may  be  stated  thus :  if  P,  Q,  E,  and  S 
represent  any  expressions  whatever,  which  contain  either  a;  or  ?/  or  both,  then 

(  P'Q=R'S,] 
the  system  of   equations  <  ^         Ms  equivalent  to  the  two  systems 

I       -P  =  "S,        J 
Q  =  R,  \  r  P  =  0,  1 

n      „    t    S'lid    ]  „  y  because  every  solution  of  either  of  the  last  two  sys- 

P  =  S,  j  [  5  =  0  ;  J 

tems  is  evidently  a  solution  of  the  first  system,  and  every  solution  of  the  first 
system  is  found  among  the  solutions  of  the  last  two  systems. 

In  Ex.  9  above,  P=x  —  2,  S  =  -7j,  Q=  x^  +  2x  +  4:,  and  R  = —  x^  +  y^. 


179] 


QUADRATIC  EQUATIONS 


307 


Ex.  10.   Solve  the  equations 


1  +  i 


13. 


Solution.   These  equations  being  fractional,  the  first  step  toward  their 
solution  would  ordinarily  be  to  clear  them  of  fractions ;  in  cases  like  this  it 

is,  however,  easier  to  regard  -   and    -  as  the  unknown  numbers,  and  to 

X  y 

eliminate  without  first  clearing  of  fractions. 

If,  for  brevity,  u  and  v  be  substituted  for  -  and  -,  respectively,  the 
given  equations  become,  respectively,  " 


and 

whence  (§  175) 

and  therefore 


3  M  -  2  y  =  3, 
m2  _|.  4  „2  =  13^ 

M  =  2  or  —  I,  and  u  =  |  or  —  |, 
a;  =  J  or  —  5,  and  2/  =  |  or  —  |. 


EXERCISES 

Solve  the  following  systems  of  equations 
x'^-\-y^  =  13, 
xy 


■■I 


6. 


12.    ^^^  +  ^^  =  1- 

25  a:y  4-  12  =  0. 

a;2  +  7/2  +  a:  =  ?/  +  26, 
xy  =  12. 

V.2  j^  „2 


13. 


14. 


15. 


16. 


17. 


r  x2  +  ?/2  =  a, 
yx-]-y  =  h. 

r  m2  4.  j,2  :=  61, 

\u  +  v  =  11. 


1  +  1  =  0. 

xy     l^ 

1      1 

a:2     ^- 

1      1 


l  +  i  =  74, 

.2  ^  „2  ' 


=  2. 


18. 


19. 


20. 


21. 


22. 


]  y    X 

[x  -y  ■- 

r  a:3  +  f 
[x  +  y  -- 


{r^  -p 
[r-p  = 

1+i: 

X^        yS 

(  x^  +  y^ 
1  x^  +  ?/^ 


16 

15' 

z2. 

=  26, 
--2. 

=  91, 

7. 


91, 


23.    \''^'^' 
[x  +  y  = 


7. 

=  2, 

=  26. 

a, 


24. 


2/ 


a:^+  ?/* 
X  +  y  = 


97, 
•  1. 


308  ELEMENTARY  ALGEBRA  [Ch.  XV 

,  rri^n^  =96-4  n,n,              '                     (  x  +  y  .  x  -y  _  10 
25.  «« + — 


26. 


J  rri'n^  =  96  - 

\m  +  n  =  6. 

(x^-\-xy-\-y^  =  8^,  -        ^ 

\x-Vry-,y  =  ^.  33.    (  ^(-^  +  ^^) 

[  X-^  +  y-l  : 


32.    \  X  -  y      X  +  y      6 
+  y^  =  45. 

:  5xy, 
1.5. 


27.    !„,,„      ,,_,o  o.      f(2  +  :r)(2/  +  l)  =  4 


.s8  -  ^8  =  37, 

s«  (s  -  f)  =  12.  34. 


(2  +  x)^-(i/  +  l)i  =  i. 


28. 


Va;  +  Vy  =  7.  35 

x'-^  —  3  a;?/  +  ?/2  =  5, 

a;4  +  V*  =  2.  36. 


I  2 Va;  +  2/  =  2Vx  -y  +  d. 


36. 
37. 


5  a;-2  _  (a;2,)-i  +  2  2/-2  z=  3. 


30. 


a;  +  y  +  2Vx  +  y  =  24,  f  3  a:^  +  3  a;?/"!  =  5^ 


—  y  +  3va;  —  y  =  10. 


■I 


3  a:.y  +  3  x-^y  =  2.5. 


31.     ,^^^  +  ^^+6V^^T?=55,  38.,_&_!.^^_,.«_5. 


|x2- 


,^  =  7.  y      X 


180.  Systems  containing  three  or  more  unknown  numbers.  Al- 
though the  solution  of  a  system  consisting  of  three  or  more 
simultaneous  quadratic  equations  (involving  as  many  unknown 
numbers  as  there  are  equations  in  the  system)  can  not  in  general 
be  made  to  depend  upon  the  solution  of  a  quadratic  equation  in 
one  unknown  number,  yet  some  solutions  of  special  cases  of  such 
systems  may  be  found  in  this  way. 

(x^+xy-hxz  =  2,  (1) 

Ex.  1.   Solve  the  equations    \  xy  -\-  y^  +  yz  =  —  2,  (2) 

[xz  +yz  +  z^  =4:.  (3) 

Solution.    Since  these  equations  may  be  written  in  the  form 

{x(x  +  y  +  z)=2,  (4) 

y(x  +  y  +  z)  =  -2,  (5) 

z(x  +  y-^z)  =  4,  (6) 

therefore,  dividing  Eqs.  (5)  and  (6)  by  Eq.  (4),  member  by  member,  we 
obtain 

^  =  -1,  and-  =  2,  (7) 

X  X 

Le.j  y  =  —  X,  and  z  =  2x\  (8) 


179-180] 


QUADRATIC  EQUATIONS 


309 


substituting  these  values  of  y  and  z,  in  terms  of  x,  Eq.  (1)  becomes 

x^  =  l, 
whence  x  =  ±l; 

and,  substituting  these  values  of  x  in  Eq.  (8),  we  obtain 

x  —  1,  y  =  —  l,  2  =  2,  and  also  x=  —  1,  y  z=l,  z  = —  2, 
as  solutions  of  the  given  system  of  equations. 

i^xy-Sx-2y  =  0, 
2xz—Sx  —  6z  =  0, 
5  3/2  +  3  y  -  4  z  =  0. 


(1) 
(2) 
(3) 


Solution.     On  dividing  these  equations  by  xy,  xz,  and  yz,  respectively, 


they  become 


y      X 
2-?-? 

Z        X 


5  +  ?_!  =  0. 

z       V 


These  last  equations,  being  of  the  first  degree  in  the  fractions  -,  -,  and 

1  1  ""  y 

~,  may  be  readily  solved  for  -,  etc.,  and  hence  the  values  of  x,  y,  and  z 
z  X 


themselves  be  found.  {  2x-\-2v  —  z  =  ^ 

Ex.  3.   Solve  the  equations  -I  x-Qy-\-z  =  2, 

[a;2-8?/2  +  3?/2=l6. 

Solution.     From  Eqs.  (1)  and  (2),  y     ^^"^ 


(1) 
(2) 
(3) 


and  z 


7  a,- -11 


4  2       ' 

substituting  these  expressions  for  y  and  z  in  Eq.  (3),  and  reducing,  it 

^^^°'^^«  5x2-12:r-9=:0, 

whence  a:  =  3  or  —  | , 

and  the  corresponding  values  of  y  and  z  are  readily  found. 


4. 


5. 


xy  =  30, 
yz  =  60, 
xz  =  50. 
a:2  +  ?/2  =  13, 
2/  +  z^  =  34, 
a-2  +  22  =  29. 


EXERCISES 


6. 


x  +  i 
xyz 

y  +  z 
xyz 

z  +  x 


=  1.2, 


=  1.5, 


310 


ELEMENTARY  ALGEBRA 


[Ch.  XV 


7. 


(z  +  x)(z  +  y)=Q. 

x^  +  2/2  +  z^  =  29, 
xy  -\-  yz  +  zx  =  -  10, 
X  +  y  +  5  =  z. 


9. 


X3/2 


^2  +  z2        5 

3' 


a;.y2 


2-^  +   X^ 

xyz 


13 

6' 


181.  Square  roots  of  binomial  quadratic  surds.  Having  now 
learned  how  to  solve  simultaneous  quadratic  equations,  it  is  pos- 
sible to  deal  with  an  interesting  problem  which  was  necessarily 
postponed  from  Chapter  XIII ;  this  problem  is  the  extraction  of 
the  square  root  of  a  binomial  quadratic  surd. 

Ex.  1.   Find  the  square  root  of  8  +  V60. 


Solution.    Let     .  Vx  +  V^  =  V  8  +  V60. 

Then,  by  squaring,      a:  +  2  Vary  +  ?/ ==  8  +  V60, 

i.e.,  x  +  y  +  2y/xy=^  +  y/m, 

whence  (§  145)  x  +  y=^  and  2Vxy  =  VOO ; 

combining  these  last  two  equations — after  squaring  the  second — easily 
leads  (§  175)  to  the  solution 

x  =  3,  ?/  =  5 ; 

therefore  Vs  +  \/60  =  V3  +  V5, 

as  is  easily  verified  by  squaring  each  member  of  this  last  equation. 

Ex.  2.   Find  the  square  root  of  a  —  V&. 
Solution.     Let  ■\/x  —  -\/y  =  V  a  —  Vb. 

Then,  as  before,  x  +  y  =  a  and  4:xy  =  b, 

whence  (§  175)  x  =  ^(a+Va^-ft)  and  y  =  i(a  -  Va^  -  &), 

and,  therefore,  ^^^~7b=  yjl+S^^H  _  ^/«Z^^, 
as  is  easily  verified. 

Note.  The  above  solution  shows  that  although  an  expression  can  always  be 
found  whose  square  is  a  —  \/b,  yet,  unless  a^  —  b  happens  to  be  a  perfect  square, 
the  expression  so  found  is  more  complicated  than  v  a  — \/&;  in  other  words,  the 
procedure  of  Exs.  1  and  2  is  of  advantage  only  when  «2  —  6  is  a  perfect  square. 


180-182]  QUADRATIC  EQUATIONS  311 

EXERCISES 

3.  In  Ex.  1  above,  why  is  x  +  y  equal  to  8,  and  2Vxy  equal  to  V60? 

Find  the  square  root  of  each  of  the  following  expressions  : 

4.  25  +  10 V6.        5.    11  +  6\/2.        6.   47  -  12vTl.         7.    18  -  6V5. 

8.  If  the  numerical  value  of  v  21  -f  SVS  is  required,  is  it  easier  to 
find  first  the  binomial  whose  square  is  21  +  8V5,  or  to  begin  by  extract- 
ing the  square  root  of  5  ?  Explain.  Also  answer  this  question  if  12  —  6  VZ 
be  substituted  for  21  +  8V5. 

182.  Square  roots  of  complex  numbers.  The  square  root  of  a 
complex  number  may  be  found  by  a  process  similar  to  that  used 
in  §  181. 

E.g.,  to  find  the  square  root  of  5  +  12  V^, 


let  Vx+ V^\/-l=  Vs  +  l-iV-l. 

Then,  by  squaring,  x-{-2  yjxy  V— 1  —  ?/  =  5  +  12  V—  1, 

whence  (§  151)  x  —  7/  =  5  and  lyjxy  =  12, 

and  therefore  (§  175)  a;  =  9  and  ?/  =  4, 

whence  V5  + 12  y/^1  =  3+2  V^, 

as  is  easily.verified. 
Similarly  in  general. 

Note.  By  means  of  extracting  square  roots  of  complex  numbers  every  imagi- 
nary number  may  be  reduced  to  the  form  a  +  6  v'—  1,  wherein  a  and  b  are  real, 
and  b^O. 

E.g.,  ^V^l=^e^=,J/3i  [</~l  =  -l 


=  Vv'-i=  Vo+V-i 

=  2  V2  -\-  2  \/2-\/—  1.  [As  in  above  example 

Similarly  in  general ;  for,  by  definition,  a  number  is  imaginary  only  when  it 
contains  an  expression  of  the  type  V—  1,  wherein  7i  is  an  even  positive  integer; 
moreover,  if  n  contains  any  odd  factors,  let  their  product  be  p  and  let  the  other 
factor  of  n  be  2* ;  then 


V^  =  ^^=l=  V-^=:i  =  2^::ri;  [p  being  odd,  </=!=- 1 

but,  by  repeatedly  extracting  the  square  root  of  an  imaginary  number  as  above, 

the  expression  V—  1  may  be  brought  to  the  form  a  +  by/—  1,  and  thus  the  given 
number  may  also  be  brought  to  this  form. 


312  ELEMENTARY  ALGEBRA  [Ch.  XV 

EXERCISES 
Find  the  square  root  of  each  of  the  following  expressions : 


1.  5  _  6^/Zri,  3.   3  +  2V-  10. 

2.  6V^^-17.  4.   5.125  -  3.75 V^^. 

5.   Reduce  v^—  1  to  an  equivalent  expression  of  the  form  a-tbV—l. 

PROBLEMS 

1.  The  sum  of  two  numbers  is  14,  and  the  difference  of  their  squares 
is  28.    What  are  the  numbers? 

2.  Find  two  numbers  whose  difference  is  15,  and  such  that  if  the 
greater  be  diminished  by  12,  and  the  smaller  increased  by  12,  the  sum  of 
the  squares  of  the  results  will  be  261. 

3.  Find  two  numbers  whose  difference  is  80,  and  the  sum  of  whose 
square  roots  is  10. 

4.  The  sum  of  two  numbers,  their  product,  and  also  the  difference  of 
their  squares,  are  all  equal;  find  the  numbers. 

5.  Find  two  numbers  whose  product  is  8  greater  than  twice  their 
sum,  and  48  less  than  the  sum  of  their  squares. 

6.  If  5  times  the  sum  of  the  digits  of  a  certain  two-digit  number  be 
subtracted  from  the  number,  its  digits  will  be  interchanged,  and  if  the 
number  be  multiplied  by  the  sum  of  its  digits,  the  product  will  be  648. 
What  is  the  number  ? 

7.  Find  two  numbers  such  that  the  square  of  either  of  them  equals 
112  diminished  by  12  times  the  other. 

8.  If  the  length  of  the  diagonal  of  a  rectangular  field,  containing 
30  acres,  is  100  rods,  how  many  rods  of  fence  will  be  required  to  inclose 
the  field? 

9.  Find  the  dimensions  of  a  rectangular  field  whose  perimeter  is 
188  rods,  and  whose  area  will  remain  unchanged  if  the  length  be  dimin- 
ished by  4  rods  and  the  width  increased  by  2  rods. 

10.  The  combined  capacity  of  two  cubical  coal  bins  is  2728  cu.  ft.,  and 
the  sum  of  their  lengths  is  22  ft. ;  find  the  length  of  the  diagonal  of  the 
smaller  bin. 

11.  It  took  a  number  of  men  as  many  days  to  pave  a  sidewalk  as  there 
were  men,  but  had  there  been  three  more  workmen  employed  the  work 
would  have  been  done  in  4  days.     How  many  men  were  employed  ? 


182]  qUABBATlC  EQUATIONS  313 

12.  A  farmer  found  that  he  could  buy  16  more  sheep  than  cows  for 
f  100,  and  that  the  cost  of  3  cows  was  $15  greater  than  the  cost  of  12 
sheep.     What  was  the  price  of  each  V 

13.  If  5  be  added  to  the  numerator  and  subtracted  from  the  denomi- 
nator of  a  certain  fraction,  the  result  will  be  the  reciprocal  of  the  fraction  ; 
and  if  2  be  subtracted  from  the  numerator,  the  result  will  be  I  of  the 
original  fraction.     What  is  the  fraction? 

14.  A  sum  of  money  at  interest  for  one  year  at  a  certain  rate  amounted 
to  $11,130.  If  the  rate  had  been  1%  less  and  the  principal  $100  more, 
the  amount  would  have  been  the  same.  What  was  the  principal  and 
what  the  rate  ? 

15.  A  certain  kind  of  cloth  loses  2%  in  width  and  5%  in  length  by 
shrinking.  Find  the  dimensions  of  a  rectangular  piece  of  this  cloth 
whose  shrinkage  in  perimeter  is  38  in.,  and  in  area  8.625  sq.  ft. 

16.  A  formal  rectangular  flower  garden  is  to  be  enlarged  by  a  border 
whose  uniform  width  is  10  %  of  the  length  of  the  garden.  If  the  area  of 
the  border  is  900  sq.  ft.,  and  the  width  of  the  old  garden  is  75  %  of  the 
width  of  the  new  one,  find  the  dimensions  of  the  garden  and  the  width 
of  the  border. 

17.  In  going  40  yds.  more  than  i  of  a  mile  the  fore  wheel  of  a  carriage 
revolves  24  times  more  than  the  hind  wheel,  but  if  the  circumference  of 
each  wheel  had  been  3  ft.  greater  the  fore  wheel  would  have  revolved  16 
times  more  than  the  hind  wheel.  What  is  the  circumference  of  the  hind 
wheel? 

18.  A  merchant  paid  $125  for  an  invoice  of  two  grades  of  sugar.  By 
selling  the  first  grade  for  $91,  and  the  second  for  $36,  he  gained  as  many 
per  cent  on  the  first  grade  as  he  lost  on  the  second.  How  much  did  he 
pay  for  each  grade  ? 

19.  Two  trains  start  at  the  same  time  from  stations  A  and  B, 
respectively,  and  travel  toward  each  other.  These  stations  are  320  miles 
apart,  and  it  requires,  from  the  time  the  trains  meet,  6  hr.  and  40  min. 
for  the  first  train  to  reach  B,  and  2  hr.  and  24  min.  for  the  second  to 
reach  A.     Find  the  rate  at  which  each  train  runs. 

20.  After  traveling  2  hr.,  a  railway  train  is  detained  1  hr.  by  an  acci- 
dent, after  which  it  proceeds  at  60  %  of  its  former  rate,  and  arrives  7  hr. 
and  40  min.  behind  time.  If  the  accident  had  occurred  50  miles  farther 
on,  the  train  would  have  saved  80  min.  What  was  the  entire  distance 
traveled  by  the  train  ? 


314 


ELEMENTARY  ALGEBRA 


[Ch.  XV 


21.  The  hundreds'  digit  of  a  3-digit  number  equals  the  sum  of  the 
other  two  digits,  the  square  of  the  tens'  digit  equals  the  units'  digit 
multiplied  by  the  sum  of  the  units'  and  hundreds'  digits,  and  if  594  be 
subtracted  from  the  number,  the  order  of  its  digits  will  be  reversed. 
What  is  the  number? 

22.  Find  the  dimensions  of  a  room  of  which  two  adjacent  side  walls 
and  the  floor  contain,  respectively,  26|,  20,  and  48  square  yards. 


III.     GRAPHIC   REPRESENTATION   OF  EQUATIONS 

183.   Graphs  of  quadratic  equations.      The  methods  of  §§  114- 
116  (which  should  now  be  reread)  are  manifestly  applicable  to 
Y  equations   of   any   degree   whatever, 

,      .  .  provided  only  that   these  equations 

contain  two  unknown  numbers. 

E.g.,  to  find  the  graph  of  the  equation 
4  a;  +  2/  =  a;2  4-  3^  it  is  merely  necessary  to  find 
a  sufficient  number  of  solutions  of  this  inde- 
terminate equation,  to  locate  the  points  having 
these  solutions  as  coiJrdinates,  and  then  to  con- 
nect these  successive  points  by  a  smooth  curve. 

Thus,  on  solving  the  above  equation  for  y, 
it  becomes  ?/  =  a;2  —  4  a;  +  3,  which  shows  that 

when  x  =  0,\,    2,  3,  4,  5,  •••,  -1,  -2,  -3,  ..., 

then     ?/  =  3,  0, -1,  0,  3,  8, —,     8,15,24,—; 

and  therefore  (§  115)  that  the  points  Pi  =  (0, 3), 

P^=  (1,  0),  P3-  (2,  - 1),  P^=  (3,  0),  P^=  (4,  3), 

2,  15),  Pq={—  3,  24),  •••  are  on  the  required 


(-1,  8),  P8  =  ( 


^6^(5.  8),. 
graph. 

If  these  points,  and  as  many  more  as  may  be  desired,  are  located  by  the  method 
of  §  114,  it  is  easily  seen  that  the  required  graph  is  approximately  represented  by 
the  curved  line  P^P^Pq  in  the  above  figure. 

If  the  above  equation  is  written  in  the  form  y  =  {x  —  1)  {x  —  S),  it  shows  that 
as  X  increases  from  3  to  00,  or  decreases  from  1  to 
—  X,  ?/  increases  from  0  to  00,  and  that  y  is  negative 
only  for  values  of  x  between  1  and  3,  i.e.,  y  is  nega- 
tive when  l<a:<3.    And  if  the  equation  is  solved 


for  X,  i.e.,  written  in  the  form 


±Vl  +  ?/,  it 


shows  that  there  are  no  points  on  the  graph  for 
which  ?/  <  —  1. 

Again,  let  the  graph  of  the  equation  4a;2  +  9?/2  =  36 
be  required.     If  this  equation  is  solved   for  y,  it 
becomes  ^  =  ±1^9— a;2^  which  shows  that  y  is  real  for  all  values  of  x  from 
«  =  — 3  to  a;  =  3,  but  imaginary  for  all  other  values  of  x,  i.e.,  this  form  of  the 


182-184] 


QUADRATIC  EQUATIONS 


315 


equation  shows  that  no  part  of  the  graph  lies  at  the  left  of  x  =  —  3,  nor  at  the 

right  of  a;  =  3.    It  also  shows  that 

when  x  =  -3,        -2,        -1,      0,  .1,  2,  and  3, 

then  7j=      0,  ±|V5,  ±t\/2,  ±2,  ±|v^,  ifVS,  and  0. 

If  the  points  having  these  solutions  as  coordinates  be  located  (§  114)  and  con- 
nected in  succession  by  a  smooth  curve  (using  approximate  values  for  the  square 
roots  indicated  above),  this  curve  will  represent  the  required  graph.  See  accom- 
panying figure. 

EXERCISES 

Construct  the  graphs  of  the  following  equations  (cf.  footnote,  p.  190): 

1.  2/2  =  8  a:.  4.   3x2-43/2  =  12. 

2.  16  a;2  +  ^2  ^  64.  5.   4  a;2  +  54  y  =  8  x  +  9  3/2  +  113. 

3.  3  x2  +  4  3/2  =  12.  6.   4  3/2  =  x^. 

7.  Show,  from  its  equation,  that  no  part  of  the  graph  of  Ex.  1  lies  to 
the  left  of  the  3/-axis  (the  line  Y'Y). 

8.  Show,  from  its  equation,  that  1.0  part  of  the  graph  of  Ex.  2  lies 
outside  of  a  certain  rectangle  whose  length  is  16  and  whose  width  is  4. 

9.  Show  from  the  equation  of  Ex.  4  that  its  graph  consists  of  four 
infinitely  long  branches,  one  in  each  of  the  quarters  into  which  the  axes 
divide  the  plane,  and  that  no  part  of  it  lies  between  x  =  —  2  and  x  =  2. 

10.  Construct  the  graph  of  the  equation  4:  x  -\-  y  =  x^  -\-  6,  and  show 
that  it  is  the  same  as  that  given  in  the  first  figure  of  §  183,  except  that 
it  is  moved  two  divisions  upward.     Explain  why  this  should  be  so. 

184.  Graphic  solution  of  simultaneous  equations.  If  the  graph 
of  one  of  two  simultaneous  equa- 
tions is  drawn  across  the  graph 
of  the  other,  i.e.,  if  the  same  axes 
are  used  for  Ijoth  graphs,  then  the 
coordinates  of  each  of  the  points  of 
intersection  of  the  two  graphs  (these 
coordinates  may  be  measured)  consti- 
tute a  simultaneous  solution  of  the 
given  equations  (cf.  §  116). 

E.g.,  the  graph  ofSx  —  5?/=—  3,  viz,  AB, 
intersects  the  graph  of  ^  x  +  y  =  z^  +  3,  viz. 
HSK,  in  the  points  P  and  Q.  The  coordinates 
of  Q,  on  being  measured,  are  found  to  be  4  and 
3,  and  those  of  P  are  approximately  ^|  and 
II ;  and  it  is  easily  verified  that  each  of  these  pairs  of  numbers  constitutes  an 
approximate  simultaneous  solution  of  the  given  equations. 


\  j 

y 

\ 

\ 

Ik 

\ 

■ 

r 

^^ 

V 

\y^ 

r 

X'    ,   ,      ,  j 

r.j 

/_ 

^ 

y^ 

V 

A 

Y 

816 


ELEMENTARY  ALGEBRA 


[Ch.   XV 


Remark.  It  should  be  observed  that  the  longei*  the  unit  divisions  on  the  axes 
are  made,  i.e.,  the  larger  the  scale  on  which  the  drawing  is  made,  the  greater 
the  degree  of  accuracy  with  which  the  coordinates  of  any  given  point  can  be 
measured. 

EXERCISES 

By  constructing  their  graphs,  find  the  approximate  simultaneous  solu- 
tions of  each  of  the  following  pairs  of  equations,  and  check  the  correct- 
ness of  your  results  by  the  methods  of  §§  174-180 : 


2. 


(9z^  +  9i/  =  289, 
I  4  x2  -  9  ?/2  =  36. 
9  a;2  +  64  2^2  _  575, 

xy  =  11. 


x^  +  9x 

y  =  -2. 

(x^-\-9x 


?/  +  7;r2-f  1, 


2/  +  7  a:2  +  1, 


185.  Graphic  solution  of  equations  containing  but  one  unknown 
number.  Manifestly  the  roots  of  the  equation  oj^  —  2a-  —  2  =  0 
are  the  values  of  x  found  by  solving  the  pair  of  simultaneous 
equations  (y  =  x'~2x-2, 

[y  =  0. 

Now,  by  §  184,  the  solutions  of  this  pair  of  simultaneous  equa- 
tions are  the  coordinates  of  the  points  in  which  their  graphs  inter- 
sect each  other,  and,  since  the  graph  of  y  =  0  is  the  line  X'X, 
therefore  the  roots  of  a;-  — 2a;  — 2=0  may  be  found  graphically 
by  measuring  the  distances  from  0  to  the  points  in  which  the 
graph  of    y  =  x'—2x—2    intersects  the  line  X'X. 

Thus,  the  graph  of  the  equation  y  =  x^  —  2x  —  2is  the  curve  MQS  in  the  figure, 
and  the  distances  OR  and  OP  are  found  to  be  approximately  2.75  and  —  .75 ; 

hence  the  roots  of  the  equation  z^  —  2x  —  2  =  0 
are  approximately  2.75  and  —  .75. 

Note  1.  Although  the  nieaswement  of  a 
root,  OR  for  example,  gives  only  a  roughly 
approximate  result,  yet,  assuming  that  the 
graph  is  continuous,  which  it  really  is,  it  is 
possible  to  find  that  result  to  any  required 
degree  of  accuracy.  Thus,  by  trial,  it  is  found 
that  7j  is  negative  when  0,  1,  and  2  are  substi- 
tuted for  X,  but  positive  when  a;  =  3 ;  therefore 
the  graph  crosses  the  line  X'X  between  x  =  2 
and  x  =  3,  i.e.,  2<OR<3.  Again,  by  sub- 
stituting 2.1,  2.2,  2.3,  •••,  for  x,  it  is  found  that 
2.7  <  0/?<  2.8;  similarly  that  2.73  <Oi?<2.74, 
2.732<Oi2<  2.733,  etc. 


184-185]  QUADRATIC  EQUATIONS  317 

Note  2.  Although  a  quadratic  equation  is  used  to  illustrate  the  method  for 
the  graphic  solution  of  numerical  equations,  yet  it  is  only  for  equations  ahove 
the  second  degree  that  this  method  is  advantageous,  —  first  and  second  degree 
equations  can  be  more  easily  solved  by  other  methods. 


EXERCISES 

1.  Show  that  one  root  oi  x^  —  7  x^  -i-  9  x  =  1  lies  between  1  and  2. 

2.  By  the  above  method  find,  correct  to  two  decimal  places,  the  root 
referred  to  in  Ex.  1. 

3.  Between  what  two  integers  do  each  of  the  other  two  roots  of  the 
equation  in  Ex.  1  lie  ?     Compare  §  184,  Ex.  4. 

4.  Corresponding  to  any  given  value  of  x,  how  does  the  value  of  y  in 
y  =  x^  —  Qx-\-Q  compare  with  its  value  in  y  =  x~  —  Q  x  +  7'^  Could, 
then,  the  graph  of  the  second  equation  be  obtained  by  merely  moving 
that  of  the  first  vertically  upward  through  one  division? 

5.  Compare  the  graphs  oi  y  =  2  x'^  —  10  a:  —  3  and  y  =  2  x^  —  10  x+l] 
also  those  oi  y  =  •}  +  4:X  —  x^  and  y  =  10  -\-  i  x  —  x^. 

6.  By  first  constructing  the  graphs  oi  y  =  x"^  —  6x  +  6,  y  =  x^  —  Qx+7, 
etc.,  compare  the  roots  of  a;2-Ga:  + 6  =  0,  x^-6x+7  =  0,  x^-Qx-\-8  =  0, 
a;2  _  6  X  +  9  =  0,  a;2  -  6  a:  +  10  =  0,  and  a;2  -  6  a:  +  11  =  0. 

7.  As  in  Ex.  6,  compare  the  two  smaller  roots  oi  x^  —  7  x^  +  9  x  —  1  =  0 
with  those  oix^-7x^-}-9x-S  =  0  and  x^  -  7  x"^  +  9  x  -  o  =  0. 

[Exercises  6  and  7  illustrate  how,  by  changing  the  absolute  term  in 
an  equation,  a  pair  of  unequal  roots  can  be  made  gradually  to  become 
equal  and  then  imaginary.] 

By  means  of  graphs  show  how  the  following  expressions  vary  in  value 
as  X  varies  gradually  from  —  go  through  0  to  +  co  : 

8.   a;2  -  7 a:  +  12.  9.   Q  +  ix  -  x^  10.   x^  -  18 a:  +  2. 


CHAPTER   XVI 

RATIO,  PROPORTION,  AND  VARIATION 

I.   RATIO 

186.  Definitions.  The  ratio  of  one  of  two  numbers  to  the  other 
is  the  quotient  obtained  by  dividing  the  first  of  these  numbers  by 
the  second.  These  numbers  themselves  are  usually  called  the 
terms  of  the  ratio,  the  first  being  the  antecedent,  and  the  second 
the  consequent. 

E.g.,  the  ratio  of  15  to  5  is  15 h- 5,  i.e.,  3;  the  ratio  of  6  to  9  is  6^-9,  i.e.,  f ; 
and  the  ratio  of  a  to  6  (whatever  the  numbers  represented  by  a  and  b)  is  a-^b. 
The  terms  of  this  last  ratio  are  a  and  b,  of  which  a  is  the  antecedent  and  6  the 
consequent. 

Each  of  the  expressions  a-i-b,  a:  b,  and  -  is  used  to  denote  the 

b 

ratio  of  a  to  b,  and  they  may  each  be  read  "  the  ratio  of  a  to  &  "  or 
"  a  divided  by  6." 

The  inverse  ratio  of  a  to  6  is  6  -=-  a,  i.e.,  it  is  the  reciprocal  of  the 
direct  ratio  of  these  numbers. 

EXERCISES 

1.  What  is  the  ratio  of  6  to  2?  of  15  to  3?  of  12  to  18?  of  4.9  to  .7? 
off  to  if? 

2.  Read  the  expression  18  :  32,  and  tell  what  it  means.  What  is  the 
inverse  ratio  of  18  to  32  ? 

3.  Write  two  other  expressions  which  mean  the  same  as  25  :  40. 

4.  Does  the  antecedent  of  a  ratio  correspond  to  dividend  or  to  divisor? 
In  the  ratio  5:8  what  is  the  antecedent?  What  is  the  other  number 
called? 

5.  What  is  meant  by  the  reciprocal  of  a  number  ?  Show  that  the 
inverse  ratio  of  x  to  y  is  the  direct  ratio  of  the  reciprocal  of  x  to  the 
reciprocal  of  y. 

6.  If  the  ratio  of  a:  to  5  equals  2.  find  x,  and  verify  your  work. 

318 


186-187]       RATIO,   PROPORTION,  AND   VARIATION  319 

7.  If  the  ratio  of  two  numbers  is  |,  and  the  consequent  is  6,  what  is 
the  antecedent  ? 

Find  X  in  each  of  the  following  ratios,  and  verify  your  result : 

8.  x^:2  =  ?y  10.   25  :  a:2  =  9. 

9.  X  :  6  =  X  -  10.  11.   36  :  a:   =  X. 

12.  Given  x^  +  6  ^^  _  5  ^y,  flnd  the  two  values  of  the  ratio  x  :  y. 

13.  The  ratio  of  two  numbers  is  f ,  and  the  ratio  which  their  sum  bears 
to  the  difference  of  their  squares  equals  that  of  1  to  7.  Find  these  num- 
bers and  verify  your  result. 

14.  Prove  that  the  value  of  a  ratio  is  not  changed  by  multiplying  or 
by  dividing  each  of  its  terms  by  any  number  whatever,  except  zero. 

15.  If  the  antecedent  of  a  ratio  be  multiplied  by  any  number,  what 
effect  will  this  have  upon  the  value  of  the  ratio  ?  Why  ?  What  is  the 
effect  of  multiplying  the  consequent?    Why  ? 

16.  Prove  that  a  ratio  which  is  less  than  1  is  increased,  and  that  a 
ratio  which  is  greater  than  1  is  diminished,  by  adding  the  same  positive 
number  to  each  of  its  terms  (cf.  §  117,  and  Ex.  17,  p.  200). 

17.  What  number  must  be  added  to  each  term  of  the  ratio  15  :  27  in 
order  that  the  resulting  ratio  shall  be  2  :  3  ?  Has  this  addition  increased 
or  diminished  the  given  ratio  ? 

187.  Ratio  of  like  quantities.  Commensurable  and  incommen- 
surable numbers.  If  A  =  7i  >  B,  "where  A  and  B  are  any  two  quan- 
tities of  the  same  kind,  and  n  is  a  number,  then  the  quantity  A  is 
said  to  have  the  ratio  n  to  the  quantity  B. 

E.g.,  since  a  line  10  inches  long  equals  2  times  a  line  5  inches  long,  therefore 
the  ratio  of  a  10-inch  line  to  a  5-inch  line  is  2,  i.e.,  it  is  the  same  as  the  ratio  of  the 
numbers  10 : 5. 

Similarly  the  ratio  of  S 6  to  $  9  is  the  same  as  6 : 9,  i.e.,  as  2 : 3. 

Since,  by  the  above  definition,  the  ratio  of  any  two  like  quanti- 
ties is  the  same  as  that  of  the  numbers  which  represent  these 
quantities,  therefore  it  is  sufficient  for  present  purposes  to  study 
the  ratios  of  numbers  only. 

If  the  ratio  of  two  numbers  (or  quantities)  is  a  rational  num- 
ber (§  130),  then  the  given  numbers  (or  quantities)  are  said  to  be 
commensurable  *  with  each  other,  but  if  this  ratio  is  an  irrational 
number,  then  they  are  said  to  be  incommensurable  with  each  other. 

*  In  this  case  the  uumbers  have  a  common  measure,  hence  the  name. 


820  ELEMENTARY  ALGEBRA  [Ch.  XVI 

E.g.,  since  VB  :  3  is  an  irrational  number,  tlierefore  y/b  and  3  are  incom- 
mensurable with  each  other;  the  diagonal  and  a  side  of  a  square  are  incommen- 
surable with  each  other,  their  ratio  being  V2  (§  130) ;  but  the  two  irrational 
numbers  3  ■\/2  and  6\/2  are  commensurable  with  each  other,  since  their  ratio  is  3 : 5. 

Note.  An  irrational  number  is  also  often  called  an  incommensurable  number, 
since  it  is  incommensurable  with  the  unit  1. 


EXERCISES 

1.  Show  that  the  following  ratios  are  all  equal :  8  bu.  oats  :  6  bu.  oats ; 
4  tons  of  coal :  3  tons  of  coal ;  1 12  :  $  9 ;  10  qt.  of  milk  :  7^  qt.  of  milk ; 
4:3;  and  ^  :  i^. 

2.  Find  the  value  of  each  of  the  following  ratios : 

8:6;  32  lb. :  4  lb. ;  4V3  in.  :  3V2  in. ;  2.7:9;  9:2.7;  4v^:V2; 
4\/2  :  2 ;     8.46  cm.  :  2.35  cm. ;     and  ^  5.80 :  29  cents. 

3.  Which  of  the  pairs  of  numbers  (or  quantities)  in  Ex.  2  are  com- 
mensurable with  each  other?     Which  are  incommensurable?     Why? 

4.  Which  of  the  individual  terms  in  Ex.  2  are  irrational? 

II.  PROPORTION 

188.  Definitions.  An  expression  of  the  equality  of  two  or  more 
ratios  is  called  a  proportion. 

E.g.,  if  a:  6  equals  c:  d,  then  the  equation  a  :  6  =  c  :  d  is  a  proportion,  and  the 
numbers  a,  6,  c,  and  d  are  said  to  he  proportional  {a\^o  in  proportion) ;  thus,  since 
6  :  3  =  10  :  5,  therefore  the  numbers  6,  3,  10,  and  5  are  in  proportion. 

The  proportion  a:  b  =  c:d  is  sometimes  written  in  the  form 
a:b  :  :c:  d,  which  is  read  "  a  is  to  &  as  c  is  to  d." 

E.g.,  the  proportion  6  :  3  : :  10  :  5  is  read  "  6  is  to  3  as  10  is  to  5  "  ;  its  meaning 
is  the  same  as  6  :  3  =  10  :  5,  i.e.,  the  same  as  S  =  ¥• 

The  first  and  fourth  terms  of  a  proportion  are  called  the  ex- 
tremes, while  the  second  and  third  terms  are  called  the  means, 
and  the  fourth  term  is  called  the  fourth  proportional  to  the  other 
three.  The  antecedents  and  consequents  of  a  proportion  are  the 
antecedents  and  consequents  of  its  two  ratios. 

E.g.,  in  the  proportion  a:b  =  c:d,  the  extremes  are  a  and d ;  the  means,  6  and 
c;  the  antecedents,  a  and  c ;  the  consequents,  6  and  d ;  and  the  fourth  proportional 
to  a,  b,  and  c  is  d. 

If  the  first  of  three  numbers  is  to  the  second  as  the  second  is  to 
the  third,  then  the  second  is  said  to  be  a  mean  proportional  between 


187-189]        RATIO,   PROPORTION,   AND    VARIATION  321 

the  other  two,  and  the  third  is  called  the  third  proportional  to  the 
other  two. 

E.g.,  in  the  proportion  a:b=^b:c  the  number  6  is  a  mean  proportional  between 
a  and  c,  and  c  is  the  third  proportional  to  a  and  b. 

A  succession  of  equal  ratios  in  which  the  consequent  of  each  is 
also  the  antecedent  of  the  next,  is  called  a  continued  proportion. 

E.g.,  it  a:b  =  b: c  =  c:d=  d:e=  '•-,  then  this  expression  is  a  continued  pro- 
portion. 

EXERCISES 

1.  Is  it  true  that  8: 12::  10:  15?  Why?  How  is  this  proportion  read  ? 
What  does  it  mean  ? 

2.  Is  it  true  that  8  :  10  :  :  12  :  15  ?  What  are  the  means,  and  what  the 
extremes,  of  this  proportion  ?  What  is  the  fourth  proportional  to  8,  10, 
and  12 ?     What  are  the  antecedents?     What  are  the  consequents  ? 

3.  How  does  the  proportion  in  Ex.  1  compare  with  that  in  Ex.  2?  If 
any  four  numbers  are  in  proportion,  will  they  be  in  proportion  after  the 
means  have  been  interchanged?  Try  several  numerical  cases,  and  also 
compare  §  189,  Prin.  5. 

4.  Show  that  the  numbers  3,  4,  6,  and  8  are  proportional  in  the  order 
in  which  they  now  stand.  Arrange  these  numbers  in  three  other  ways 
in  each  of  which  they  will  be  proportional. 

5.  Show  that  6  is  a  mean  proportional  between  4  and  9 ;  also  between 
2  and  18.  Is  —  6  also  a  mean  proportional  between  these  numbers? 
What  are  the  third  proportionals  in  these  cases? 

189.  Important  principles  of  proportion.  Since  a  proportion  is 
merely  an  equation  whose  members  are  fractio7is,  the  principles  of 
proportion  may  be  easily  derived  (as  is  shown  below)  from  those 
already  demonstrated  for  equations  and  fractions.  » 

Principle  1.  If  four  numbers  are  in  proportion,  then  the 
product  of  the  means  equals  the  product  of  the  extremes* 


*  Before  reading  the  proofs  of  these  principles  the  student  is  urged  to  make 
several  numerical  illustrations  of  each,  and  also  to  try  to  make  a  general  proof 
for  himself,  which  he  may  then  compare  with  that  given  in  the  text.  Verbal 
statements  of  these  principles  should  be  committed  to  memory. 

If  the  terms  of  a  proportion  are  quantities,  they  may  first  be  replaced  by  their 
representative  numbers  (cf .  §  187) ,  after  which  the  above  principle  may  be  applied ; 
the  product  of  two  quantities  is  meaningless. 


322  ELEMENTARY  ALGEBRA  [Ch.  XVI 

For,  let  a,  b,  c,  and  d  be  any  four  numbers  which  are  in  propor- 
tion, then  ^  .  5  ^  c  :  d ; 

a_c 
b     d 

whence  ad  =  be,  [Multiplying  by  bd 

which  was  to  be  proved. 

Principle  2.  If  the  product  of  two  nurrbbers  equals  the 
product  of  two  others,  then  these  four  numbers  form  a  pro- 
portion of  which  the  two  factors  of  either  product  may 
be  made  the  means,  and  those  of  the  other  product  the 
extremes.* 

For,  if  ad  =  be, 

then  ^  =  ^ ,  [Dividing  by  bd 

i.e.,  a:b  =  c:d. 

In  the  same  way  it  may  be  shown  that,  if  ad  =  be,  then 

b  :  a  =  d  :  c,   e:  a  =  d:b,  etc. ; 

hence  the  correctness  of  Principle  2. 

Remark.  From  the  proof  just  given  it  follows  that  the  correct- 
ness of  a  proportion  is  established  when  it  is  shown  that  the  product 
of  the  means  equals  the  product  of  the  extremes;  this  test  is  very 
useful. 

Principle  3.  Tlie  products  of  the  corresponding  terms  of 
two  {or  more)  proportions  are  proportional. 

For,  if  a:b  =  e:d  and  e  \f=g  :  h, 


then  (multiplying)   «.!  =  £. |,,>.,p  =  ^^, 

hence  ae  :  bf=  eg  :  dh, 

which  was  to  be  proved. 


Principle  2  is  the  converse  of  Principle  1. 


189]  RATIO,  PROPORTION,  AND   VARIATION  323 

Principle  4.  Tlie  quotients  of  the  corresponding  terms 
of  two  proportions  are  proportional. 

For,  if  a:h=^c:d  and  e  :  /=  g  :  h, 

then  ad  =  be  and  eh  =fg, 

whence  adfg  =  bceh ;  [§24  (2) 

on  dividing  each  member  of  this  last  equation  by  ehfg,  it  becomes 

ad _  be    .       a    d _b    c 
eh      fg'   '  ''  e    h     f   g' 

and  from  this  last  equation,  by  Principle  2, 

a  .  b_  c  ,  d 
e'f~g'h' 
which  was  to  be  proved. 

Principle  5.    If  a:b  =  c:d, 

then  (1)  b:a  =  d:e', 

(2)  a:c  =b:d', 

(3)  {a  +  b):  a(or  b)  =  (c-\-d):  c(or  d)-, 

(4)  (a  —  b):  a{or  b)  =  (c  —  d):  c{or  d) ; 
and                (5)  {a-\-b)  :  (a-b)  =  (c  +  d)  :  (c-d). 

The  correctness  of  these  proportions  [(1)  to  (5)]  easily  follows 
from  the  remark  at  the  end  of  Principle  2;  the  detailed  proofs 
are  left  as  an  exercise  for  the  student. 

Eemark.  Proportion  (1),  above,  is  usually  said  to  be  formed 
from  the  given  proportion  by  inversion ;  (2)  by  alternation  ;  (3)  by 
composition ;  (4)  by  division  (or  by  separation) ;  and  (6)  by  compo- 
sition and  division. 

The  student  should  translate  each  part  of  the  above  principle  into 
verbal  language,  and  commit  it  to  memory ;  e.g.,  (3)  thus  translated 
is  :  If  four  numbers  are  in  proportion,  then  they  are  also  in  propor- 
tion when  taken  by  composition;  i.e.,  the  sum  of  the  first  and  second 
is  to  the  first  (or  the  second)  as  the  sum  of  the  third  and  fourth  is  to 
the  third  (or  the  fourth). 


324  ELEMENTARY  ALGEBRA  [Ch.  XVI 

Principle  6.  In  a  series  of  equal  ratios  the  sum  of  the 
antecedents  is  to  the  sum  of  the  consequents  as  any  ante- 
cedent  is  to  its  own  consequent. 

Thus,  if  a:b  =  c:d  =  e:f=g:h=  "'  =  .^:y, 

then    (a  +  c  +  e  +  9'+  •••  +»):  (6  +  d +/+/i  +  •-.  -\-y)=:e:f. 

To  prove  this  theorem,  let  each  of  the  given  equal  ratios  be 
represented  by  a  single  letter,  say  r ; 

then  l=r,  ^=r,  ^  =  r,  f  =  r,  ...,  and  ?=  r, 

b  d         f         h  y 

hence         a  =  hr,  c  =  dr,  e  =  fr,  g  =  hr,  •  •  •,  and  x  —  yr, 
and,  adding  these  equations,  member  to  member, 

a  +  c  +  e-l-gr+  ...  ^x=(b-\-d+f+h+  "- +  y)r, 
and  therefore      ^  +  c +6  +  ^  +  -  +  o^ ^^e 

which  proves  the  principle. 

Note.  As  in  the  proof  just  given,  so  it  will  often  be  found  advantageous  to 
represent  a  ratio  by  a  single  letter. 

Principle  7.  Lihe  powers  of  proportional  numbers  are 
proportional;  so  also  are  like  roots;  i.e.,  if 

a:b  =  c:  d,  then  a"* :  6**  =  c'* :  c?". 

For,  if  ^  =  ^,  then  f^X  =  f^X,  i.e.,  ^-  =  ^; 

'  b     d'  \bj       \dj  '         b^     d"' 

hence,  if  a:b  =  c:d,  then  a~ :  6**  =  c"  :  c^",* 

which  was  to  be  proved. 

EXERCISES 

1.  Find  the  fourth  term  of  the  proportion  of  which  the  first  three 
terms  are  5,  12,  and  15. 

Suggestion.     Let  x  represent  the  fourth  term,  and  apply  Principle  1. 

2.  Find  a  mean  proportional  between  4  and  25.  How  many  answers 
has  this  problem  ? 

3.  Find  the  third  proportional  to  25  and  40. 

*  According  as  n  is  an  integer  or  its  reciprocal,  a»»  is  a  power  or  a  root  of  a. 


189]  EATIO,   PROPORTION,  AND   VARIATION  326 

4.  If  a  line  18  inches  long  is  divided  into  two  parts  whose  ratio  is 

4 : 5,  how  long  is  each  part  ? 

5.  If  x:15=(a;-l):12,finda:. 

6.  If  32  :  x^  =  iL  :  (a;  +  2),  find  x. 

7.  Find  the  mean  proportionals  between  am^  and  a^m ;  also  between 
a  +  b  and  a  —  b. 

8.  li  a  :  b  =  c  :  d,  show  that  am  :bn  =  cm:  dn,  wherein  m  and  n  are 
any  numbers  whatever ;  also  translate  this  principle  into  verbal  language. 

9.  Show  that  the  product  of  the  means  of  a  proportion,  divided  by 
either  extreme,  equals  the  other  extreme. 

10.  Show  that  the  mean  proportional  between  any  two  numbers  is 
the  square  root  of  the  product  of  these  numbers. 

11.  Prove  Principle  6  by  means  of  the  remark  under  Principle  2. 

12.  Prove  Principle  4  by  using  a  single  letter  to  represent  a  ratio 
(compare  proof  of  Principle  6). 

13.  Add  1  to  each  member  of  the  equation  a:h  =  c  :d,  write  the  result 
in  the  form  of  a  proportion,  and  thus  prove  (3)  of  Principle  5. 

14.  It  a:b  =  c  :  d,  and  if  a  is  not  equal  to  b  nor  to  c,  show  that  no  num- 
ber whatever  can  be  added  to  each  term  of  the  proportion  and  leave  the 
results  in  proportion. 

If  p  :  q  =  r  :  s,  prove  that : 

15.  r  :  .9  =  -  :  -  •  17.   pr  :  as  =  r^  :  s^. 

q  p 

16.  5p:dr=5q:Ss.  18.   (p  -]-  q)  :  (r  +  s)  =  Vp^  +  (/^  :  Vr^  +  s^. 

xy.  v.iven  ^^y^2x):(rj-2x)  =  (12x  +  6y~Sy.iQy-12x-l)^^ 
find  x  and  y. 

20.  Given  x  :27  =  y  :  9  =  2  :  (x  -  j/);  find  x  and  y. 

21.  li  a  :  b  =  c  :  d  -  e  :f  =  g  :  h  =  •",  and  I,  m,  n,  p,  •••  are  any  numbers 
whatever,  prove  that 

(jna  +  Ic  —  ne  +  pg  +  •••)  :  (mb-j-  Id  —  nf  +  ph  -\-  "•)  =  a:h. 

22.  If  a  :  X  =  b  :  y  =  c  :  z  =  d  :  w  =  ••• ,  show  that 

(a«  +  6"  +  c»  +  ...)  :  (x~  +?/'»  +  2« ...)  =  a"  .  x\ 

23.  If  (p  +  q  -h  r  -h  s)  (p  -  q  -  r  +  s)  =  (p  -  q  +  r  -  s)  (p  -\-  q  -  r  -  s), 
show  that p  :  q  =  r  :s. 


326  ELEMENTARY  ALGEBRA  [Ch.  XVI 

24.   It  a:b=  c  :d  =  e  :f,  show  that 


c:d  =  Va2  +  4  ac  +  5  c2  :  Vft2  +  4  6c?  +  5  c?2 
25.   If  (x  —  y)  :  (y  —  2)  :  (2  —  x)  =  / :  m  :  n*,  and  x  ^y  =^z,  show  that 
/  +  w  +  n  =  0. 

By  the  principles  of  proportion,  solve  the  following  equations : 

2g^   Vx  +  7  +  Vx  ^4:+  Vx 
Vx  +  7  —  Vx      4  —  Vx 
Suggestion.    Apply  Principle  5  (6). 

27    x+  VT^n:  ^  IS 


a;  _  V.r  -  1        7 

28.  (a  -  V2  ax  -  a;^)  :  («  _  6)  =  (a  +  V2  ax  -  x^)  :  (a  +  6). 
Suggestion.    First  apply  Principle  5  (2). 

29.  If  qx±^^ay±_cz^azj^^^  ^^^^  ^^^^  ^^^^  ^^  ^j^^^^   ^^^.^^ 

&y  +  c?2     62  +  c?x     6x  +  <iy 

equals  ^-^-^ 


b  +  d 

30.  The  perimeter  of  a  triangle,  whose  sides  are  in  the  ratio  5  :  6  :  8,  is 
57  meters ;  find  the  lengths  of  the  sides. 

31.  Divide  16  into  two  parts  such  that  their  product  is  to  the  sum  of 
their  squares  as  3  :  10. 

32.  Find  two  integers  whose  ratio  is  the  same  as  15f  :  9|.  Can  the 
ratio  ot.any  two  numbers  whatever  be  expressed  by  means  of  two  inte- 
gers (cf .  Ex.  2,  p.  320)  ? 

33.  By  the  addition  of  new  books,  a  certain  circulating  library  was 
increased  in  the  ratio  of  12  :  11 ;  later  160  old  books  were  discarded,  and 
it  was  then  found  that  the  library  remained  increased  only  in  the  ratio 
35  :  33.     How  many  books  were  there  in  the  library  originally? 

34.  If  x,  y,  and  z  represent  positive  numbers,  which  of  the  following 

ratios  is  the  greater,   2^±A^  or  ^±1^?    ^  "  -^  +  "  or  £±X±^? 
2x  +  7y         x  +  Sy      x  +  y  —  z         x  —  y  —  z 

35.  If  a :  &,  c:d,  e:f,  g:h,  •••  are  unequal  ratios,  in  which  a,  b,  c,  ••• 
are  positive  numbers,  and  if  a  :  ft  is  the  greatest  and  e  :/the  least  among 
these  ratios,  show  that  (a  +  c  +  e-{-g  +  •.•)  :  (b  -\-  d  +/+  h  +  ...)  is  less 
than  a  :  b,  but  greater  than  e  :/. 

•  *  The  expression  a:h:  c  =x:y:z,  means  that  a:b  =  z:y,  a:  c  =  z:z,  and 
& :  c  =  y : «.    It  may  also  be  written  a:x  =  b:y  =  c:z. 


189-190]       RATIO,   PROPORTION,  AND    VARIATION  327 


III.   VARIATION 

190.  Definitions.  Many  questions  in  mathematics  are  con- 
cerned with  numbers  whose  values  are  changing;  such  numbers 
are  usually  spoken  of  briefly  as  variables,  while  numbers  whose 
values  do  not  change  are  called  constants. 

Two  variables  may,  also,  be  so  related  that  a  change  in  one  of 
them  necessarily  produces  a  corresponding  change  in  the  other. 

E.g.,  if  w  and  v  represent,  respectively,  the  weight  and  volume  (i.e.,  the  number 
of  pounds,  and  the  number  of  cubic  feet)  of  the  quantity  of  water  in  a  certain 
tank,  and  if  a  cubic  foot  of  water  weighs  62.5  pounds,  then  w  =  62.5  v. 

Moreover,  while  the  water  is  flowing  into  this  tank,  both  w  and  v  will  mani- 
festly change  (i.e.,  they  will  be  variables),  but  through  all  their  changes  the 
relation  between  these  variables  continues  to  be 

w  =  62.5  V. 

Of  two  variables  which  are  so  related  that,  during  all  their 
changes,  their  ratio  remains  constant,  each  is  said  to  vary  as 
the  other.* 

E.g.,  if  X  and  y  are  any  two  variables  so  related  that,  during  all  their  changes, 
x:y  =  k,  wherein  ^  is  a  constant,  then  x  varies  as  y,  and  y  also  varies  as  x. 

The  equation  x:y  =  k,oT,  what  is  the  same  thing,  x  =  ky,  shows  that  when  y 
is  doubled,  tripled,  halved,  etc.,  then  x  is  also  doubled,  tripled,  halved,  etc. 

The  symbol  employed  to  denote  variation  is  oc;  it  stands  for 
the  words  "  varies  as,''  and  the  expression  a  ccb  is  read  "  a  varies 
as  6." 

In  the  above  example  about  the  water,  vj  varies  as  v,  because  their  ratio  is 
constant  {i.e.,  w.v  =  62.5,  whatever  the  quantity  of  the  water)  ;  this  is  com- 
monly expressed  by  saying  that  "the  weight  of  water  varies  as  its  volume." 

One  of  two  numbers  is  said  to  vary  inversely  as  the  other  if  the 
ratio  of  the  first  to  the  reciprocal  of  the  second  is  constant. 

E.g.,  the  time  required  for  a  railway  train  to  travel  a  given  route  varies 
inversely  as  its  speed;  for,  if  t,  r,  and  d  represent,  respectively,  the  time,  rate, 
and  distance,  then 

t'r=  d,  that  is,  t:-=d, 

where  d  is  constant.  From  the  first  of  these  equations  it  follows  also  that  if  the 
speed  is  doubled,  then  the  time  will  be  halved;  if  the  speed  is  divided  by  3,  then 
the  time  will  be  trebled,  etc. 


♦Also  " to  vary  directly  as  the  other.* 


328  ELEMENTABY  ALGEBRA  [Cii.  XVI 

Again,  if  x,  y,  and  z  are  variables  such  that  x  =  kyz,  where  k  is 

a  constant,  then  x  is  said  to  vary  jointly  as  y  and  Z]  and  if  x  =  — , 
then  X  is  said  to  vary  directly  as-  y  and  inversely  as  z. 

Note.    It  should  be  remarked  in  passing  that  such  an  expression  SiS  w  xv 

above  (i.e.,  the  weight  of  water  varies  as  its  volume)  is  merely  an  abbreviated 

form  of  the  proportion 

w:w  =v:v  , 

wherein  w  and  w'  stand  for  the  respective  weights,  and  v  and  v'  for  the  volumes, 
of  any  two  quantities  of  water. 

The  theory  of  variation  is  substantially  included  in  that  of  ratio  and  propor- 
tion, and  the  only  reason  for  even  defining  the  expressions  "  varies  as,"  "  varies 
inversely  as,"  etc.,  here,  is  that  this  convenient  phraseology  is  so  well  established 
in  physics,  chemistry,  etc. 

EXERCISES  AND  PROBLEMS 

1.  Which  of  the  following  quantities  are  constants  and  which  are 
variables:  (1)  the  circumference  of  a  growing  orange?  (2)  the  length 
of  the  shadow  cast  by  a  certain  church  steeple  between  sunrise  and  sun- 
set? (3)  the  length  of  the  steeple  itself?  (4)  the  time  since  the  dis- 
covery of  America  ?  (5)  the  interest  earned  by  a  note  ?  (6)  the  principal 
of  the  note  ? 

2.  What  is  meant  by  the  expression,  "  the  speed  being  constant,  the 
distance  traveled  by  a  railway  train  varies  as  the  time "  ?  Express  this 
fact  by  means  of  a  proportion  (cf.  note,  above). 

3.  What  is  meant  by  saying  "  the  interest  earned  by  a  certain  princi- 
pal varies  as  the  time "  ?  Express  this  fact  as  a  proportion ;  also  as  an 
equation. 

4.  What  is  meant  by  the  expression  x  <x  y?  Are  a;  and  y  constants  or 
variables  here? 

5.  Express  by  means  of  an  equation  that  x  cc  y.     Explain. 

6.  If  a:  cc  y,  and  if  x  =  12  when  ?/  =  3,  find  the  equation  connecting 
X  and  ?/,  and  the  value  of  x  when  y  =  7. 

Solution.  Since  cc  act/,  therefore  cc  =  ^t/,  where  ^  is  a  constant.  Moreover, 
if  a;  =  12  when  2/  =  3,  then  the  equation  x  =  ky  gives  12  =  3  A;,  from  which  we 
find  A;  =  4  ;  therefore,  under  the  given  conditions,  z  =  ^y;  and  therefore  cc  =  28 
when  y  =  7. 

7.  If  a;  varies  inversely  as  y,  and  a:  =  10  when  y  =  3,  what  is  the 
value  of  X  when  y  =  5  ? 

8.  If  m  varies  inversely  as  n,  and  is  equal  to  4  when  n  =  2,  what  is 
the  valne  of  r?  when  w  =  li? 


190]  RATIO,  PBOPOliTION,  AND   VARIATION  329 

9.  The  area  of  a  circle  varies  as  the  square  of  its  radius,  and  the 
area  of  a  circle  whose  radius  is  10  ft.  is  314.6  sq.  ft.  What  is  the  area 
of  a  circle  whose  radius  is  5  ft.  ?  of  one  whose  radius  is  12  ft.  ? 

10.  Find  the  radius  of  a  circle  whose  area  is  twice  as  great  as  that  of 
a  circle  whose  radius  is  10  ft.  (cf .  Ex.  9) . 

11.  If  one  of  two  numbers  varies  inversely  as  the  other,  show  that 
their  product  is  constant. 

12.  li  AazB  and  BcxzC,  prove  that  AxC. 
Suggestion.    Show  that  A  =  kC,  wherein  k  is  some  constant. 

13.  Ji  mccn  and  pccn,  prove  that  m  ±  pccn. 

14.  If  p  varies  inversely  as  q  and  q  varies  inversely  as  r,  prove  that 
peer. 

15.  If  3  w^  —  18  cc  2  n  +  1,  and  m  =  4  when  n  =  2,  what  is  the  value 
of  m  when  n  =  23.5  ? 

16.  If  X  varies  as  y  when  z  is  constant,  and  as  z  when  y  is  constant, 
prove  that,  when  both  y  and  z  vary,  xcc  yz;  i.e.,  that  x  varies  jointly  as  y 
and  z. 

Suggestion.  Let  y  and  z  vary  separately,  and  write  each  variation  as  a  pro- 
portion; thus  from  the  change  in  y,  -^=  -^,  and  now  letting  z  change,  ^=  ^> 


z  _  yz 


X-     y 


whence  — -  =  ^7— ,  from  which  the  conclusion  is  evident. 
X       y  z 

17.  The  area  of  a  triangle  varies  as  its  altitude  if  its  base  is  constant, 
and  as  its  base  if  its  altitude  is  constant.  If  the  area  of  a  triangle  whose 
base  and  altitude  are,  respectively,  6  and  5  in.,  is  15  sq.  in.,  what  is  the 
area  when  the  base  and  altitude  are  13  and  10  in.  respectively? 

18.  If  the  volume  of  a  pyramid  varies  jointly  as  its  base  and  altitude, 
and  if  the  volume  is  20  cu.  in.  when  the  base  is  12  sq.  in.  and  the  altitude 
is  5  in.,  what  is  the  altitude  of  the  pyramid  whose  base  is  48  sq,  in.  and 
whose  volume  is  76  cu.  in.  ? 

19.  The  distance  (in  feet)  fallen  by  a  body  from  a  position  of  rest 
varies  as  the  square  of  the  time  (in  seconds)  during  which  it  faUs.  If 
a  body  falls  257|  ft.  in  4  sec,  how  far  will  it  fall  in  5  sec.  ?  how  far 
during  the  5th  second?  how  far  during  the  7th  second? 

20.  If  the  intensity  of  light  varies  inversely  as  the  square  of  the  dis- 
tance from  its  source,  how  much  farther  from  a  lamp  must  a  book,  which 
is  now  2  ft.  away,  be  removed  so  as  to  receive  just  one  third  as  much 
light? 


330  ELEMENTARY  ALGEBUA  [Ch.  XVI 

21.  A  rectangle  moves  with  its  center  on  a  given  straight  line  and  its 
plane  perpendicular  to  that  line.  If  one  of  its  sides  varies  as  the  dis- 
tance, and  an  adjacent  side  as  the  square  of  the  distance,  of  the  rectangle 
from  a  certain  point  on  this  line,  and  if  at  the  distance  3  ft.  the  rectangle 
becomes  a  square  2  ft.  on  a  side,  what  is  its  area  when  the  distance  is 
5ft.? 

22.  In  order  that  two  weights  attached  to  a  rod  should  balance  each 
other  when  the  support  on  which  the  rod  rests  is  between  them,  their 
distances  from  the  point  of  support  should  vary  inversely  as  the  weights. 
Find  the  point  of  support  for  a  12-foot  plank  on  which  two  boys  weigh- 
ing 75  and  90  lb.,  respectively,  wish  to  play  see-saw. 

23.  The  number  of  oscillations  made  by  a  pendulum  in  a  given  time 
varies  inversely  as  the  square  root  of  its  length.  If  a  pendulum  39.1 
inches  long  oscillates  once  a  second,  what  is  the  length  of  a  pendulum 
that  oscillates  twice  a  second  ? 

24.  The  volume  of  a  sphere  varies  as  the  cube  of  its  radius,  and  the 
volume  of  a  sphere  whose  radius  is  1  ft.  is  4.19  cu.  ft.  Find  the  volume 
of  a  sphere  whose  radius  is  3  ft. 

25.  Three  metal  spheres  whose  radii  are  3,  4,  and  5  in.  respectively, 
are  melted  and  formed  into  a  single  sphere.  Find  the  radius  of  this 
new  sphere. 

Suggestion.  If  S^  and  S^  are  the  volumes  of  two  spheres  whose  radii  are 
rj  and  r^,  and  if  -S  is  a  sphere  of  radius  r  and  equivalent  to  S^  -i-  S^,  then  Si  =  kr^, 
and  S  =  Si  -1-  Sa  =  A  {r^  +  r^^)  =  kr». 


CHAPTER    XVII 
SERIES  -  THE  PROGRESSIONS 

191.  Definitions.  A  series  is  a  succession  of  related  numbers 
which  conform  to  some  definite  law.  The  numbers  which  con- 
stitute the  series  are  called  its  terms. 

The  law  of  a  series  may  prescribe  the  way  each  of  the  terms, 
after  a  given  term,  is  formed  from  those  which  precede  it,  or  it 
may  state  how  each  term  is  related  to  the  number  of  the  place  it 
occupies  in  the  series. 

E.g.,  in  the  series  1,  2,  3,  5,  8,  13,  •••  eacli  term,  after  the  second,  is  the  sum  of 
the  two  preceding  terms. 

In  the  series  2,  6,  18,  54,  •••  each  term,  after  the  first,  is  3  times  the  preceding 
term;  and  3,  7,  11,  15,  19,  •••  is  a  series  of  which  each  term,  after  the  first,  is 
formed  by  adding  4  to  the  preceding  term. 

On  the  other  hand,  in  the  series  1,  4,  9, 16,  25,  •••  each  term  is  the  square  of  the 
number  of  its  place  in  the  series;  and  the  law  of  the  series  §,  |,  f,  f,  tt»  •" 
is  expressed  by  ,  where  n  is  the  number  of  the  term's  place  in  the  series. 

1  ~T"  ^  M 

If  the  number  of  terms  of  a  series  is  unlimited,  it  is  called  an 
infinite  series,  otherwise  it  is  a  finite  series. 

E.g.,  in  each  of  the  five  examples  given  above  the  series  is  infinite,  but  the 
series  1,  2,  3,  5,  8,  13,  •••  89  is  finite,  consisting  of  10  terms. 

Only  the  simplest  kinds  of  series  —  the  so-called  "progres- 
sions "  —  will  be  studied  in  the  present  chapter. 

I.   ARITHMETICAL  PROGRESSION 

192.  Definitions  and  notation.  A  series  in  which  the  difiierence 
found  by  subtracting  any  term  from  the  next  following  term  is 
the  same  throughout  the  series  is  an  arithmetical  series  ;  it  is  also 
often  called  an  arithmetical  progression,  and  is  designated  by 
"A.  P."  This  constant  difference,  which  may  be  either  positive 
or  negative,  is  called  the  common  difference  of  the  series. 

331 


332  ELEMENTARY  ALGEBRA  [Ch.  XVII 

E.g.,  the  numbers  2, 5, 8, 11, 14,  •••  form  an  A.  P.  because  5  —  2  =  8  —  5  =  11  —  8 
=  14  —  11  =  ••• ;  the  common  difference  of  this  series  is  3. 

So,  too,  18,  11,  4,  —  3,  — 10,  •••  is  an  A.  P.  whose  common  difference  is  —  7. 

In  any  given  A.  P.  it  is  customary  to  represent  the  first  term, 
the  last  term,  the  common  difference,  the  number  of  terms,  and 
the  sum  of  all  the  terms,  by  the  letters  a,  I,  d,  n,  and  s,  respec- 
tively ;  and  these  are  called  the  elements  of  the  series. 

E.g.,  in  the  series  2,  5, 8,  •••  32,  the  elements  are :  a  =  2, 1  =  32,  d  =  3,  n  =  11,  and 
s  =  187. 

EXERCISES 

1.  Define  a  series.  If  a  row  of  numbers  be  written  down  quite  at 
random,  will  they  constitute  a  series  ?     Explain. 

2.  Define  an  arithmetical  series.     Is  1,  7,  13,  19,  25,  an  A.  P.  ?    What 

are  its  elements? 

3.  If  the  series  7, 11, 15, 19  be  continued  toward  the  right,  what  is  the 
next  term?  Why?  Extend  this  series  by  writing  the  next  four  terms 
at  the  right,  and  also  the  next  three  at  the  left. 

4.  Do  the  numbers  7,  11,  and  15  belong  to  the  same  A.  P.  as  27,  31, 
and  35?  What  is  d  for  each  of  these  two  series?  Write  the  series 
which  includes  both  of  these  sets  of  numbers. 

5.  If  the  first,  third,  and  fifth  terms  of  an  A.  P.  are  18,  24,  and  30, 
respectively,  find  d  and  write  8  consecutive  terms  of  this  series.  Also 
write  10  consecutive  terms  of  the  series  of  which  19,  9,  and  4  are  the 
first,  fifth,  and  seventh  terms,  respectively.    What  is  d  for  this  last  series? 

6.  Are  the  numbers  5,  5  +  3,  5  +  6,  5  +  9,  and  5  +  12  an  A.  P.?  What 
are  the  values  of  a,  d,  I,  n,  and  s  for  this  series?  How  may  the  second 
term  of  this  series  be  fonned  from  the  first?  the  third  from  the  second? 
any  terra  from  the  one  preceding  ? 

7.  Are  the  numbers  x,  x  -{■  y,  x  -{■  2 y,  x  -\-  Sy,  a;  +  4y,  •••  an  A. P.? 
Why?  What  is  d  in  this  series?  How  may  the  second  term  be  formed 
from  the  first?  the  third  from  the  first?  the  fourth  from  the  first?  the 
tenth  from  the  first?  the  fifteenth  from  the  first?  How  may  any  term 
whatever  (say  the  nth)  be  formed  from  the  first? 

8.  Show  from  the  definition  of  an  A.  P.  that  such  a  series  may  be 
written  in  the  form 

a,  a  -\-  d,  a  -{-  2  d,  a-j-Zd,  •■-  I  -  2  d,  I  -  d,  I, 

wherein  a,  d,  and  /  represent,  respectively,  the  first  term,  common  differ- 
ence, and  last  term . 


192-193]  SERIES  —  THE  PROGRESSIONS  333 

193.  Formulas.  The  elements  of  an  A.  P.  are  connected  by  two 
fundamental  equations  (formulas),  which  will  now  be  derived. 

Since  each  term  of  an  A.  P.  may  be  derived  by  adding  d  to  the 
preceding  term  (cf.  Exs.  6-8,  §  192),  therefore,  if  I  stands  for  the 

^*^*^^^'  l  =  a+(7i-l)d.  (1) 

Again,  since  the  sum  of  the  terms  of  an  A.  P.  may  be  written 
in  each  of  the  two  following  forms, 

s  =  a  +  {a-{-d)  -{-  {a  -{-2 d)  +  '"  +  {1-2 d)  -\-  (I- d)  +  1, 

and     s  =  l-j-{l-d)-{-{l-2d)-\ f-  (a  +  2 d)  +  («  +  ^)  4- «, 

therefore,  by  adding  these  equations,  term  by  term, 

2s  =  (a  +  Z)  +  (a  +  ^)4-(a+/)+-H-(a4-0  +  («  +  0  +  («  +  0; 
i.e.,  2s  =  n(a-\-l),  [n  terms 

whence  g  =  M«ilil.  '         (2) 

Note.  If  any  three  of  the  five  elements  of  an  A.  P.  are  given,  the  other  two 
can  always  be  found  from  formulas  (1)  and  (2)  above,  because,  in  that  case,  the 
remaining  two  unknown  elements  will  be  connected  by  two  independent  equations 
(cf.  Ex.  17,  p.  334). 

EXERCISES  AND  PROBLEMS 

1.  Verify  formulas  (1)  and  (2)  above,  in  the  case  of  the  arithmetical 
series  7,  10,  13,  16,  19,  22,  25.  What  is  the  value  of  a  in  this  series? 
of  d?    of  n?  of  Z? 

2.  Verify  formulas  (1)  and  (2)  above,  for  the  arithmetical  series 
26,  19,  12,  5,  -  2,  -  9,  -  16,  -  23,  -  30 ;  also  for  the  series  -  8,  -  5|, 
-  3i  -  1,  H,  3i  6,  8i  lOf,  13. 

3.  By  means  of  formula  (1)  find  the  17th  term  of  7,  11,  15,  ••• ;  then, 
using  formula  (2),  and  without  writing  all  the  terms,  find  the  sum  of  the 
first  17  terms  of  this  series. 

4.  Using  formulas  (1)  and  (2)  find  the  8th  term,  and  also  the  sum 
of  the  first  8  terms  of  1,  3.5,  6,  8.5,  •-. 

5.  Find  the  26th  term,  and  also  the  sum  of  the  first  18  terms  of  the 
series  1,  5,  9,  — . 

6.  Find  the  sum  of  10  terms  of  4,  11,  18,  •••. 

7.  Find  the  sum  of  30  terms  of  -  2,  -  0.5,  1,  2.5,  •... 


334  ELEMENTARY  ALGEBRA  [Ch.  XVII 

8.  Find  the  sum  of  19  terms  of  2,  5,  8,  ••• ;   also  find  the  sum  of 
k  terms  of  this  series. 

9.  Find  the  sum  of  n  terms  of  the  series  5,  5  +  /:,  5  +  2  ^,  5  +  3  ^,  •••. 

10.  Find  the  sum  of  t  terms  of  the  series  h,  2h,^h,  •••.  What  is  this 
sum  if  ^  =  2  and  f  =  50  ? 

11.  Find  the  sum  of  the  even  numbers  from  2  to  100  inclusive. 
Compare  your  result  with  that  found  in  Ex.  10. 

12.  How  many  strokes  does  a  clock  make  during  the  24  hours  of 
a  day? 

13.  Suppose  that  50  eggs  were  placed  in  a  row,  each  2  yds.  from  the 
next,  and  a  basket  2  yds.  beyond  the  last  eg^,  how  far  would  a  boy, 
starting  at  the  basket,  walk  in  picking  up  these  eggs  and  carrying  them, 
one  at  a  time,  to  the  basket  ? 

14.  If  a  body  falls  16.1  feet  during  the  first  second,  3  times  as  far  during 
the  next  second,  5  times  as  far  during  the  third  second,  etc.,  how  far  will 
it  fall  during  the  8th  second?   how  far  during  the  first  8  seconds? 

15.  If  the  6th  and  11th  terms  of  an  A.  P.  are,  respectively,  17  and  32, 
find  the  common  difference,  and  also  the  sum  of  the  first  11  terms. 

Suggestion.  Since  the  6th  term  is  17,  therefore  17=a  +  5d.  Similarly, 
32  =  a  + 10  d.    From  these  two  equations  find  a  and  d,  and  then  find  s. 

16.  By  means  of  formula  (1)  find  the  number  of  the  terms  in  the 
series  2,  6,  10,  •••,  66.     Also  find  the  sum  of  the  series. 

17.  How  many  terms  are  there  in  the  series  —  1,  2,  5,  •••  if  the  sum  of 
this  series  is  221  ? 

Suggestion.  Since  in  this  series  a  =  —  1,  d?  =  3,  and  s  =  221,  therefore  formulas 
(1)  and  (2)  of  §  193  become,  respectively,  Z  =—  1  +  (n  -  1)  3  and  221  =  -  (—  1  +  0 ; 
and  from  these  equations  it  is  easy  to  determine  n  and  I. 

18.  Determine  the  unknown  elements  in  the  series  •••,  10,  13,  16,  •••  if 
s  =  112  and  n  =  7. 

19.  If  s,  71,  and  d  are  given,  find  a  and  I,  i.e.,  find  a  and  I  in  terms  of 
s,  n,  and  d  (cf.  Ex.  18). 

20.  Find  a  and  n  in  terms  of  c?,  Z,  and  s.  Make  up  and  solve  eight 
other  examples  of  this  kind. 

21.  Show  that  an  A.  P.  is  fully  determined  when  any  three  of  its 
elements  are  given. 

22.  Prove  that  the  products  obtained  by  multiplying  each  term  of  an 
A.  P.  by  any  given  number  are  themselves  in  arithmetical  progression. 

If  each  term  of  an  A.  P.  be  divided  by  any  given  number,  or  be  in- 
creased or  diminished  by  any  given  number,  will  the  results  be  in  arith- 
metical progression?    Explain. 


193-194]  SERIES — THE   PROGRESSIONS  335 

194.  Arithmetical  means.  The  two  end  terms  of  an  arithmetical 
series  are  called  the  extremes  of  the  series,  while  all  the  other 
terms  are  called  the  arithmetical  means  between  these  two. 

E.g.,  in  the  series  5,  9, 13,  17,  21,  the  extremes  are  5  and  21,  and  9,  13,  and  17 
are  arithmetical  means  between  5  and  21. 

Ex.  1.   Insert  5  arithmetical  means  between  3  and  27. 

Solution.  Since  there  are  to  be  5  means  between  3  and  27,  therefore 
the  complete  series  will  consist  of  7  terms,  and  therefore,  for  this  series, 
a  =  3,  Z  =  27,  and  n  =  7  -,  whence,  from  formula  (1)  of  §  193,  d  =  i,  and 
the  series  is :  3,  7,  11,  15,  19,  23,  27. 

EXERCISES  AND   PROBLEMS 

2.  Insert  4  arithmetical  means  between  12  and  27. 

3.  Insert  15  arithmetical  means  between  19  and  131. 

4.  Insert  20  arithmetical  means  between  16  and  —  40. 

5.  If  m  arithmetical  means  are  inserted  between  two  given  numbers, 
such  as  a  and  b,  show  that  the  common  difference  for  the  series  thus 
formed  is  d  =  (h  —  a)-^  (m  +  1). 

6.  If  X  is  the  (one)  arithmetical  mean  between  a  and  b,  show,  directly 
from  the  definition  of  an  A.  P.,  that  x  =(a+  b)-^2.  Does  this  agree 
with  the  statement  in  Ex.  5?    Explain. 

7.  Without  actually  finding  the  means  asked  for  in  Ex.  2,  find  the 
sum  of  the  series  formed  by  inserting  them. 

8.  Find  3  numbers  in  A.  P.  whose  sura  is  15  and  the  sum  of  whose 
squares  is  107. 

Suggestion.    Let  x  —  y,x,  and  x  +  y  represent  the  required  numbers. 

9.  The  sum  of  7  terms  of  an  A.  P.  is  105,  and  the  sum  of  its  third  and 
fifth  terms  is  10  times  its  first  term.     Find  the  series. 

10.  The  product  of  the  extremes  of  an  A.  P.  of  3  terms  is  4  less  than 
the  square  of  the  mean,  and  the  sum  of  the  series  is  24.     Find  the  series. 

11.  The  sum  of  4  numbers  in  A.  P.  is  14,  and  the  product  of  the  means 
is  12.     What  are  the  numbers  ? 

Suggestion.    Let  x  —  Sy,  x  —  y,  x  +  y,  and  x  +  3y  represent  the  series. 

12.  The  sum  of  an  A.  P.  of  5  terms  is  15,  and  the  product  of  the  ex- 
tremes is  3  less  than  that  of  the  second  and  fourth  terms.    Find  the  series. 

13.  How  many  arithmetical  means  must  be  inserted  between  4  and 
25  so  that  the  sum  of  the  series  may  be  116? 


336  ELEMENTARY  ALGEBRA  [Ch.  XVII 

14.  A  number  consists  of  3  digits  which  are  in  A.  P. ;  and  the  sum  of 
the  digits  multiplied  by  30.4  equals  the  number,  but  if  9  be  added  to  the 
number,  the  units'  and  tens'  digits  will  be  interchanged.  What  is  the 
number  ? 

15.  In  the  series  1,  3,  5,  •••  what  is  the  nth  term?  Prove  that  the  sum 
of  the  first  n  odd  numbers,  beginning  with  1,  is  n^. 

II.   GEOMETRIC  PROGRESSION 

195.  Definitions  and  notation.  A  series  in  which  the  quotient 
of  any  term  (after  the  first)  divided  by  the  next  preceding  term 
is  the  same  throughout  the  series  is  a  geometric  series  ;  it  is  also 
often  called  a  geometric  progression,  and  is  designated  by  "  G.  P." 
This  constant  quotient  is  called  the  common  ratio,  or  simply  the 
ratio,  of  the  series. 

E.g.,  the  numbers  2,  6,  18,54,  •••  form  a  geometric  series,  whose  ratio  is  3; 
while  §,  —  1,  ^,  —  I,  ¥-,  •••  is  a  G.  P.  whose  ratio  is  —  f . 

It  is  customary  to  represent  the  elements  of  a  G.  P.,  i.e.,  the  first 

term,  the  last  term,  the  number  of  terms,  the  ratio,  and  the  sum 

of  all  the  terms,  by  the  letters  a,  I,  n,  r,  and  s,  respectively. 

E.g.,  in  the  G.  P.  2,  -(5,  18,  -54,  162,  -486,  1458,  a  =  2,^=1458,  n=7, 
r=— 3,  and  s=1094. 

EXERCISES 

1.  Is  7,  21,  63,  189,  567  a  geometric  series?  Why?  What  are  its 
elements  ? 

2.  Is  2,  8,  32,  96,  288  a  geometric  series?     If  not,  why  not? 

3.  Is  -  6,  12,  -  24,  48,  -  96,  192,  -  384,  768  a  G.  P.?  What  are  its 
elements  ?  How  may  the  second  term  be  obtained  from  the  first  ?  the 
third  from  the  second  ?   the  sixth  from  the  fifth  ? 

4.  If  the  series  in  Ex.  3  be  continued  toward  the  right,  what  is  the 
next  term?  the  next  after  that?  Extend  this  series  5  terms  toward 
the  left  also. 

5.  If  a  represents  the  first  term  of  a  G.  P.,  and  r  the  ratio,  what  is 
the  second  term?  the  third?  the  fourth  ?  the  fifth?  the  fourteenth?  the 
twenty-third?  the  nth  ?     Explain. 

6.  Show  that  x,  xy,  xy^,  xy%  xy'^,  ...  is  a  G.  P.  What  are  a  and  r  in 
this  series  ?  ^  _ 

Answer  these  questions  with  regard    to    ~,  p^,  p^q\  p(i\  q^,  —  also. 


194-190]  SERIES— THE   PROGUESSIONS  337 

7.  What  is  r  in  the  series  2,  |,  f,  •••  ?  in  the  series  21,  7,  |,  •••  ?  Are 
these  two  series  merely  parts  of  the  same  series?     Explain. 

8.  If  the  first,  third,  and  sixth  terms  of  a  G.  P.  are  12,  3,  and  f, 
respectively,  find  r,  and  then  write  down  the  first  8  terms  of  this  series. 

196.  Formulas.  The  elements  of  a  G.  P.  are  connected  by  two 
fundamental  equations  which  will  now  be  derived  (cf.  §  193). 

Since  each  term  of  a  G.  P.  may  be  obtained  by  multiplying  the 
preceding  term  by  r  (cf.  Exs.  5  and  6,  §  195),  therefore,  if  I  repre- 
sents the  nth  term  of  such  a  series,  then 

/  =  ar^'-K  (1) 

Again,  if  s  represents  the  sum  of  a  G.  P.  of  n  terms,  then 

s  =  a  +  a/'  +  ar^  -I-  ar^  H h  ar"~^  +  ar"~^, 

whence     sr  =  ar  +  cn^  +  a?*^  +  •  •  •  +  ar"^"^  +  ar",      [multiplying  by  r 

and  therefore,  by  subtracting  the  second  of  these  equations  from 
the  first,  member  from  member, 

s  —  sr  =  a  —  ar"", 

hence  s  = (2) 

1  —  ?' 

EXERCISES  AND   PROBLEMS 

1.  By  means  of  formula  (1)  above,  write  down  the  fifth  term  of  the 
G.P.  7,  21,  63,  .... 

2.  By  formula  (1)  write  down  the  seventh  term  of  3,  6,  12,  •••,  and 
then  find  the  sum  of  the  first  7  terms  of  this  series  by  means  of  formula 
(2).  Verify  your  answers  by  actually  writing  the  first  7  terms  of  the 
given  series. 

3.  Find  the  G.  P.  whose  third  term  is  18  and  whose  eighth  term  is 
4374. 

Suggestion.  Since  the  third  term  is  18,  therefore,  by  formula  (1),  18  =  ar^; 
similarly,  4374  =  ar^;  therefore,  by  dividing  the  second  of  these  equations  by  the 
first,  243  =  r6,  i.e.,  r  =  3;  etc. 

4.  Find  the  G.  P.  whose  fifth  term  is  f  and  whose  ninth  term  is  ^|f . 
Also  find  the  sum  of  this  series. 

5.  Find  the  sum  of  the  first  10  terms  of  the  series  1,  2,  4,  •••. 

6.  Find  the  sum  of  the  first  6  terms  of  1,  1.5,  2.25,  •.-. 


338  ELEMENTARY  ALGEBRA  [Ch.  XVll 

7.  Find  the  sum  of  the  first  7  terms  of  2,  -  |,  |,  ■••. 

8.  Find  the  sum  of  the  first  7  terms  of  1,  —  2 a:,  4 x%  •••. 

9.  Find  the  sum  of  the  first  k  terms  of  —  5,  —  2,  —  .8,  .... 

10.  Find  the  sum  of  the  first  9  terms  of  the  series  whose  first  term 
is  13.5  and  whose  fourth  term  is  4. 

11.  By  actually  dividing  a  -  ar'\  i.e.,  a (1  -  r"),  by  1  -  r,  verify  the 
correctness  of  formula  (2)  of  §  196  [cf.  §  68  (1)]. 

12.  Show  that  the  sum  of  n  terms  of  a  G.  P.  may  be  expressed  in 
each  of  the  following  forms: 

a  —  li    rl  —  a     nr"  —  «     «   ^        «  «^'* 


1  —  r'r— 1       r  —  1  1—  r      1 


r 


13.  If  r,  n,  and  I  are  given,  find  a  and  s ;  i.e.,  find  a  and  s  in  terms  of 
r,  n,  and  I  (cf.  Ex.  19,  §  193). 

14.  By  means  of  formulas  (1)  and  (2),  §  196,  show  that  a  G.  P.  is 
fully  determined  when  any  three  of  its  elements  are  given  (cf.  Ex.  21, 
§193). 

15.  If  r  =  3,  do  the  terms  of  the  series  a,  ar,  ar%  ar%  ■-  ar'^~'^  increase 
or  decrease  in  going  toward  the  right?  Can  you  name  a  number  so  large 
that  it  will  exceed  the  nth  term  of  this  series  for  all  values  of  n,  however 
large  ? 

16.  If  r  >  1  (numerically),  show  that  the  terms  of  the  series  a,  ar,  ar% 
ar%  •••  grow  larger  and  larger  in  passing  toward  the  right,  and  that,  by 
taking  n  sufficiently  large,  the  nth  term,  i.e.,  ar"'~^,  may  be  made  to  exceed 
any  given  finite  number  however  large. 

17.  If  r<  1  (numerically),  show  that  the  terms  of  the  series  a,  ar,  ar^, 
ar^,  ..•  grow  smaller  and  smaller  in  passing  toward  the  right,  and  that, 
by  taking  n  sufl&ciently  large,  ar^-'^  may  be  made  to  differ  from  zero  by 
less  than  any  given  number  however  small.* 

♦  Suggestion  on  Exs.  16  and  17.  Let  h  be  any  positive  number,  then  since 
(1  +  hy-  (1  +  hy-l  =  (1  +  hy--^{{l+h)-l} ^h{l+hy-\  and  since  hO--\-hy-^> h, 
when  5  —  1  is  positive,  therefore  {1  + h)^— (l  +  h)  >h,  (l-j-h)^— (l-\- h)^>h, 
(l  +  A)4-(l  +  /i)3>/i,  (i  +  /i)5_(i  +  /i)4>/i,  ...and  (l  +  /i)«- (1  + /i)'»-i>  A. 
Now  adding  these  inequalities,  and  the  equation  l-{-h  =  l-{-h,  member  to  mem- 
ber, we  have  {1  +  h)n  ';>  1  -{■  nh  ;  but  1  +  «^  >  Q  (where  Q  is  any  given  number 
however  large)  when  n>(Q  — l)-^A,  hence,  for  this  or  larger  values  of  n, 
(l  +  7i)">Q;  and  therefore,  by  taking  n  large  enough,  the  nth  power  of  any 
number  greater  than  1  can  be  made  to  exceed  any  number  however  large. 

Again,  letp<l  and p  .  q  =  l,then  q(=  1  -r-p)  >1,  and  therefore  q^^i.e.,  l-^p", 
may  be  made  larger  than  any  given  number  however  large,  hence  p"  may  be  made 
smaller  than  any  given  number  however  small. 


196-197]  SERIES — TEE  PROGRESSIONS  339 

18.  Three  numbers  whose  product  is  216  form  a  G.  P.,  and  the  sum 
of  their  squares  is  189'.     What  are  the  numbers? 

Suggestion.    Let  - ,  a,  and  ar  represent  the  requh'ed  numbers. 
r 

19.  If  the  population  of  the  United  States  was  76,000,000  in  1900,  and 
if  it  doubles  itself  every  25  years,  what  will  it  be  in  the  year  2000  ? 

20.  Thinking  $1  per  bushel  too  high  a  price  to  pay  for  wheat,  a 
man  bought  10  bu.,  paying  3  cents  for  the  first  bushel,  6  cents  for  the 
second,  12  cents  for  the  third,  and  so  on.  AVhat  did  the  tenth  bushel 
cost  him,  and  what  was  the  average  price  per  bushel  ? 

21.  A  gentleman  loaned  a  friend  $250  at  the  beginning  of  each  year 
for  4  years.  If  money  is  worth  5  %  compound  interest,  how  much  should 
be  paid  back  to  him  at  the  end  of  the  fourth  year  to  discharge  the 
obligation  ? 

22.  Divide  38  into  three  parts  which  are  iii  G.  P.,  and  such  that  when  1, 
2,  and  1  are  added  to  these  parts,  respectively,  the  result  shall  be  in  A.  P. 

197.  Infinite  decreasing  geometric  series.  If  r<l  (numerically), 
the  G.  P.  is  called  a  decreasing  series,  while  if  ?'  >  1  (numerically), 
it  is  an  increasing  series. 

Formula  (2)  of  §  196,  which  gives  the  sum  of  the  first  n  terms  of 
the  series  a,  ar,  ar^,  ai^,  •••  may  evidently  be  written  in  the  form 

a  ar'' 


1  —  r     1  —  ?• 

Now,  for  a  decreasing  series  the  value  of  becomes  smaller 

1  — r 

and  smaller,  and  approaches  zero  as  a  limit  when  n  becomes  in- 
finite (cf.  Ex.  17,  p.  338)  ;  therefore  the  sum  of  the  first  n  terms  of 
an  infinite  decreasing  G.  P.  may,  by  taking  n  sufficiently  large, 

be  made  to  differ  from  by  less  than  any  given  number  how- 
ever small.  This  is  usually  expressed  by  saying  that  the  sum  to 
infinity  of  a  decreasing  G.  P.  is  ^  ;  and  if  s^  stands  for  "limit 
of  «„  when  7i  becomes  infinite,"  it  may  be  written  thus : 

s   -     ^ 
1  — r 


340  ELEMENTARY  ALGEBRA  [Ch.  XVII 

EXERCISES  AND  PROBLEMS 

1.  From  a  line  one  foot  long  cut  off  one  half,  then  one  half  of  the  re- 
mainder, then  one  half  the  next  remainder,  and  so  on ;  if  this  process 
were  continued  without  end,  show  that,  when  expressed  in  inches,  the 
parts  cut  off  form  the  G.  P. : 

6,  3,  I,  I,  f,  ^,  ^2,  ii,  .-. 

2.  By  means  of  formula  (2),  §  196,  find  s^  for  the  series  in  Ex.  1.  Also 
find  Sg,  Sq,  *iq,  and  s„. 

3.  Based  upon  the  manner  in  which  the  series  in  Ex.  1  was  formed, 
show  that  5„<  12,  however  large  n  may  be.  How  near  to  12  will  s„  ap- 
proach as  n  is  made  larger  and  larger?     Explain.     Also  find  Soo  by  §  197. 

4.  Find  s„  for  the  series  0.6,  0.06,  0.006,  ••.,  and  thus  show  that  0.6, 
i.e.,  that  0.666  •••,  equals  |. 

Find  s^  for  each  of  the  following  series : 


5. 

1,  -  h  h  •••• 

9.   0.3. 

6. 

1,  h  h  •••• 

10.   0.i2. 

7. 

i  -  f,  A,  -. 

11.    1.362. 

8. 

V2,  1,  \/o:5,  •••• 

12.   4.7523. 

.5. 

If,  in  a  G.  P.,  r  is 

positive  and  le 

13.    l,k,k^ 
(wherein  k 

<1). 

14.   ^,-,K 
X    x^ 

(wherein  x 

>1)' 

the  series  is  greater  than  all  the  terms  that  follow  it. 

16.  If  a  point  moves  from  a  given  position,  and  along  a  straight  line, 
with  such  a  velocity  that  during  any  given  second  it  moves  75  %  as  far  as 
it  did  during  the  preceding  second,  and  if  it  moved  24  feet  during  the 
first  second,  how  far  will  it  move  before  it  comes  to  rest? 

17.  If  a  sled  runs  80  feet  during  the  first  second  after  reaching  the 
bottom  of  a  hill,  and  if  its  distance  decreases  20%  during  each  second 
thereafter,  how  far  will  it  run  on  the  level  before  coming  to  rest  ? 

18.  If  a  ball,  on  being  dropped  from  a  tower  window  100  feet  above 
the  pavement  rebounds  40  feet,  then  falls  and  rebounds  16  feet,  and  so 
on,  how  far  will  it  move  before  coming  to  rest? 

19.  The  president  of  a  woman's  charity  organization  starts  a  "  letter 
chain"  by  writing  3  letters,  each  numbered  1,  requesting  each  recipient 
to  remit  10  cents  to  the  society,  and  also  to  send  out  3  other  letters,  each 
numbered  2,  with  a  similar  request,  the  chain  to  close  with  the  letters 


197-198]  SERIES  —  THE  PROGRESSIONS  341 

numbered  20.     If  evefy  one  addressed  complies  with  the  requests,  how 
much  money  will  be  realized  for  the  society  ? 

20.  Although  Sao  for  the  series  ^,  \,  I,  •••  is  1,  show  that  for  the  series 
hhhh  •••>  ^n  grows  larger  beyond  all  bounds,  by  sufficiently  increasing  n. 

Suggestion.    Write  the  series  tlius:  Sn=^+(J+i)  +  (i+B+7+J)H ,  putting 

8  terms  in  the  next  group,  16  in  the  next,  and  so  on,  and  show  that  each  group  is 
greater  than  5- 

198.  Geometric  means.  The  two  end  terms  of  a  finite  G.  P.  are 
called  its  extremes,  while  all  the  other  terms  are  called  the  geo- 
metric means  between  these  two. 

E.g.,  in  the  series  |,  ^,  |,  |,  and  ^^  the  extremes  are  f  and  ^,  and  I,  I,  and  | 
are  geometric  means  between  them. 

Ex.  1.   Insert  4  geometric  means  between  f  and  —  ^. 

Solution.  Since  4  means  are  to  be  inserted,  therefore  the  complete 
series  will  consist  of  6  terms,  and  therefore,  for  this  series,  a  =  ^,  I  =  —  ^-, 
and  n  =  6;  hence,  by  formula  (1)  of  §  196, 

-  ^=  I  •  r^  therefore  r^  =  -  »-^^,  i.e.,  r  =  -  f , 

and  the  _series  is  :         I,  —  I,  1,  —  I,  f,  and  —  ^. 

EXERCISES 

2.  Insert  4  geometric  means  between  3  and  96. 

3.  Insert  3  geometric  means  between  2  and  ^\  (two  answers). 

4.  Insert  5  geometric  means  between  x^  and  y^  (two  answers). 

5.  If  m  geometric  means  are  inserted  between  any  two  given  num- 
bers, such  as  a  and  b,  show  that  the  common  ratio  for  the  series  thus 
formed  is  ""^y/b  -^  a. 

6.  If  X  is  the  (one)  geometric  mean  between  a  and  b,  show  directly 
from  the  definition  of  a  G.  P.  that  x  =  Vab.  Does  this  agree  with  the 
statement  in  Ex.  5?    Explain. 

7.  Insert  a  geometric  mean  between  12  and  3.  Give  two  solutions, 
and  compare  Ex.  6. 

8.  Insert  a  geometric  mean  between  0.5  and  3.5 ;  also  between 
(a  +  6)2  and  (a  —  b)^;   and  between  dm^x^  and  75m-^x. 

9.  If  the  difference  between  two  numbers  is  24,  and  if  their  arithmeti- 
cal mean  exceeds  their  geometric  mean  by  6,  what  are  the  numbers  ? 


342  ELEMENTARY  ALGEBRA  [Ch.  XVII 

199.  Arithmetico-geometric  series.  A  series  formed  by  multi- 
plying corresponding  pairs  of  terms  of  an  A.  P.  and  a  G.  P.  is 
usually  called  an  arithmetico-geometric  series.  The  sum  of  n  terms 
of  such  a  series  may  be  found  by  the  method  of  §  196.* 

Ex.  1.     Find  the  sum  of  the  series  1,  2  r,  3  r^,  4  r^,  5  r*,  .•••  nr''-\ 


Solution. 

Let 

s  = 

1  + 

2  r  +  3  r2  +  4 

r3  +  ... 

.  +  n/-"-i, 

then 

rs  — 

r  + 

2r2+  3r3  +  .- 

•  +  (n 

-  l)r"-i  + 

nr^, 

whence 

s  - 

-rs  — 

1  + 

r  +  r^+r^+. 

...  +  7- 

i.e., 

.(1- 

-r)  = 

1  - 

1  - 

r 

[§  196,  ] 

form 

and  therefore 

s  = 

1  - 
(1- 

-  r)2      1  -  r 
EXERCISES 

2.  By  the  method  of  Ex.  1  find  the  sum  of  the  n  terms  of  the  series 
obtained  by  multiplying  the  corresponding  terms  of  the  two  series  a, 
a  +  rf,  a  -f  2  c?,  •••  a  +  (n  -  l)d  and  1,  r,  r\  •••  r"-i. 

3.  Find  the  sum  of  the  series  whose  (n  +  l)th  term  is  (a  +  nh)x''\  i.e.,  find 
a  +(a  +  &)a;  +  (a  +  2  6)a;2+  ...  +  (a  +  nh)x'^.* 

III.    HARMONIC   PROGRESSION 

200.  Harmonic  series.  A  series  of  numbers  whose  reciprocals 
form  an  A.  P.  is  called  an  harmonic  series ;  it  is  also  often  called 
an  harmonical  progression,  and  is  designated  by  "  H.  P." 

E.g.,  the  series  1,  i,  \,  iV,  •••  is  an  H.  P.  because  the  reciprocals  of  its  terms  are 
1, 4,  7, 10,  ••.,  and  these  form  an  A.  P. 

It  follows  immediately  from  the  above  definition  that  questions 
concerning  harmonic  series,  which  admit  of  solution,!  ^ay  be 
solved  by  treating  the  reciprocals  of  the  terms  of  the  given  series 
as  an  A.  P. 

E.g.,  to  extend  the  H.P.  ?,  \,  i\,  ^ps  three  terms  further  at  each  end  it  is  only- 
necessary  to  take  the  reciprocals  of  these  numbers,  which  form  the  A,  P.  5, 4,  V.  '/> 
in  which  d  =  §,  and  extend  it  three  terms  at  each  end,  and  write  the  reciprocals  of 
its  terms.    Thus,  the  given  series  extended  is  —  §,  —  1,  f ,  ?,  \,  ^,  ^^,  i,  a'l. 

*  For  an  extension  of  this  subject  see  Chrystal's  Algebra,  Part  I,  p.  489. 
t  There  is  no  general  formula  for  the  sum  of  n  terms  of  an  H.  P. 


199-200]  SERIES  —  THE  PROGRESSIONS  343 


EXERCISES 

1.  If  X  is  the  harmonic  mean  between  a  and  b,  show,  as  above,  that 

1  _  1  =.  1  _  1,  and  hence  that  x  =  ■^^' 
X     a      0      X  a-^  b 

2.  Insert  5  harmonic  means  between  2  and  —  3. 

3.  The  arithmetical  mean  between  two  numbers  is  5,  and  their  har- 
monic mean  is  3.2.     What  are  the  numbers  ? 

4.  The  difference  between  two  numbers  is  2,  and  their  arithmetical 
mean  exceeds  their  harmonic  mean  by  |.     Find  the  numbers. 

5.  Given  (b  —  a)  :  (c  —  b)=  a:x,  prove  that  x  equals  a,  b,  or  c,  accord- 
ing as  a,  b,  and  c  form  an  A.  P.,  a  G.  P.,  or  an  H.  P. 

6.  If  the  sixth  term  of  an  H.  P.  is  ^,  and  the  seventeenth  term  is  ^j, 
find  the  thirty-seventh  term. 

7.  If  a  and  b  are  any  two  unequal  positive  numbers,  show  that  their 
arithmetical  mean  is  greater  than  their  geometric  mean,  and  that  this, 
in  turn,  is  greater  than  their  harmonic  mean ;  also  that  the  geometric 
mean  is  a  mean  proportional  between  their  arithmetical  and  harmonic 


CHAPTER  XVIII 
MATHEMATICAL  INDUCTION  —  BINOMIAL  THEOREM 

201.  Proof  by  induction.  An  elegant  and  powerful  form  of 
proof,  and  one  that  finds  extensive  application  in  almost  every 
branch  of  mathematics,  is  what  is  known  as  "proof  by  induc- 
tion." 

Suppose  it  to  have  been  found,  by  trial  or  otherwise,  that  x  —  y 
is  a  factor  of  a?  —  y^,  o?  —  y^,  and  x^  —  y^,  and  that  one  wishes  to 
know  whether  it  is  a  factor  of  x^  —  if,  x^  —  y^,  •••  also.  Actual 
trial  with  any  one  of  these,  say  x'—'if',  would  show  that  it  is  exactly 
divisible  by  x  —  y,  but,  besides  being  somewhat  tedious,  this 
division  gives  no  information  as  to  whether  ic  —  ?/  is  or  is  not  a 
factor  of  x^  —  y^,  •••  also  ;  each  successful  trial  increases  the  proh- 
dbility  of  the  success  of  the  next,  but  it  really  proves  nothing 
beyond  the  single  case  on  trial. 

That  x  —  y  is  a  factor  of  a?"  —  ?/",  for  every  positive  integral 
value  of  n,  may  be  proved  as  follows : 

Since  ic"  —  y^  =  x  (a?""^  —  2/""^)  +  y^~^  (x  —  ?/), 

therefore  x  —  y  is  sl  factor  of  .t"  —  ?/",  if  it  is  a  factor  ofx'^''^  —  ^/""K 
In  other  words :  if  x  —  y  is  a,  factor  of  the  difference  of  two  like 
integral  powers  of  x  and  y,  then  it  is  a  factor  of  the  difference  of 
the  next  higher  powers  also. 

But  since,  by  actual  trial,  x  —  y  is  already  known  to  be  a  factor 
of  x'^  —  y*,  therefore,  by  what  has  just  been  proved,  it  is  also  a 
factor  of  a^  —  y^',  again,  since  it  is  now  known  to  be  a  factor  of 
a^  —  y^,  therefore  it  is  a  factor  of  x^  —  y^  -,  and  so  on  without  end : 
i.e.,  x  —  y  is  a  factor  of  a;**  —  ?/"  for  every  positive  integral  value 
of  n  [cf.  §  68  (i)]. 

The  proof  just  given  is  an  example  of  what  is  known  as  a 
proof  by  mathematical  induction ;  it  consists  essentially  of  two 
steps,  viz. : 

344 


201]  MATHEMATICAL  INDUCTION  345 

(a)  Showing,  by  trial  or  otherwise,  the  correctness  of  a  given 
proposition   when   applied  to  one  or  more  particular  cases,   and 

(h)  Proving  that  if  the  proposition  is  true  for  any  given  case, 
then  it  is  true  for  the  next  higher  case  also. 

From  (a)  and  (6)  it  then  follows  that  the  proposition  under 
consideration  is  true  for  all  like  cases.* 

EXERCISES 

1.  Prove  that  the  sum  of  the  first  n  odd  integers  is  n^. 
Solution,     (a)  By  trial  it  is  found  that  1  +  3  =  22  and  1  +  3  +  5  =  32. 

(6)   Moreover,  t/  1  +  3  + 5 H \-{2k-l)  =  k2,  (1) 

then,  by  adding  the  next  odd  integer  to  each  member  of  Eq.  (1),  we  have 
l  +  3  +  5  +  ...  +  (2A;-l)  +  (2A;  +  l)  =  A;2+(2A;  +  l)  =  (A:  +  l)2; 
i.e.,  if  the  law  in  question  is  true  for  the  first  k  odd  integers,  then  it  is  true  for 
the  first  k  +  1  odd  integers  also. 

But,  by  actual  trial,  this  law  is  known  to  be  true  for  the  first  3  odd  integers, 
hence  it  is  true  for  the  first  4;  and,  since  it  is  7iow  known  to  be  true  for  the  first 
4,  therefore  it  is  true  for  the  first  5,  and  so  on  without  end ;  hence  the  sum  of  any 
number  of  consecutive  odd  integers  beginning  with  1  equals  the  square  of  that 
number. 

By  matliematical  induction  prove  that : 

2.  1  +  2  +  3  +  ...  +  n  =  -1  n  (n  +  1). 

3.  2 +  4  + 6  +  •••  + 2w  =  n(n  + 1). 

4.  12  +  22  +  32  +  ...  +  7i2  =  1  n  (n  +  1)  (2  n  +  1). 

5.  13  +  28+33+  ...  +n8  =  in2(n  + 1)2^(1  +  2  +  3  +  ...  +  n)2. 

6.  A:  +  ;r^+:r^  +  .-.+  ^ 


1-2      2.3      3.4  n(n+l)      n  +  1 

7.  1.2  +  2.3  +  3.4  +  ...  +  n(n+l)  =  in(n  +  l)  (n+ 2). 

8.  Having  established  (a)  and  (b)  in  the  inductive  proof  of  any  prop- 
osition, show  the  generality  of  the  proposition  by  showing  that  there 
can  be  no  first  exception,  and  therefore  no  exception  whatever. 

*  The  student  should  carefully  distinguish  between  mathematical  induction , 
as  here  defined,  and  what  is  known  as  inductive  reasoning  in  the  natural 
sciences;  a  proof  by  mathematical  induction  is,  from  its  very  nature,  ahsolutely 
conclusive.  On  the  other  hand,  the  inductive  method  in  physics,  chemistry,  etc., 
consists  in  formulating  a  statement  of  a  law  which  will  fit  the  particular  cases 
that  are  known,  and  regarding  it  as  a  law  only  so  long  as  it  is  not  contradicted 
by  other  facts,  not  previously  taken  into  account.  From  the  nature  of  the  case 
step  (&)  above  can  not  be  applied  in  physics,  etc. 


346  ELEMENTARY  ALGEBRA  [Ch.  XVIII 

202.  The  binomial  theorem.  The  method  of  induction  (§  201) 
furnishes  a  convenient  proof  of  what  is  known  as  the  binomial 
theorem;  this  theorem,  which  was  presented  without  formal  proof 
in  §  62,  may  be  symbolically  stated  thus : 

(a;  +  yy  =  a;"  +  nx^-'y  +  ^  ^^  7^^)  x^^-y 

wherein  x-\-y  represents  any  binomial  whatever,  and  n  is  any 
positive  integer. 

To  prove  this  theorem  by  mathematical  induction,  observe  first 
that  it  is  correct  when  n  =  2,  for  it  then  becomes 

2  •  1 

(x  -i- yy  =  3(^ -\- 2  xy  i-  — —  xy,  i.e.,  (x  +  yy  =  x^-{-2xy  +  y^, 

which  agrees  with  the  result  of  actual  multiplication. 

Again,  if  Eq.  (1)  is  true  for  any  particular  value  of  n,  say  for 
n  =  k,  i.e.,  if 

(x-{-yy=x^-^7c^-'y-^ ^^^ x'^-y+  ^^^~^^%~^^  ^"V  +  •  •  •,    (2) 
then,  on  multiplying  each  member  of  Eq.  (2)  hj  x-\-y,  it  becomes 
(x+yy^'=x'^^^+kx'y-^^^^^^a^-y+^^^ 

1  •  ^ 

i.e.,  (x+yy+'=x'-^'  +  (k+l)x'y+^\~^'^^^ x'-y 

+  (±i^l|fclla--y+..,  (3) 

*  The  student  should  now  re-read  §  62,  and  observe  that  the  second  member  of 
this  identity  conforms  in  every  detail  to  the  statement  there  given. 


202-203]  BINOMIAL   THEOREM  347 

which  is  of  precisely  the  same  form  as  Eq.  (2),*  merely  having 
A:  +  1  wherever  Eq.  (2)  has  k.  Moreover,  Eq.  (3)  is  obtained  from 
Eq.  (2)  by  actual  multiplication,  and  is  therefore  true  if  Eq.  (2) 
is  true ;  hence,  if  the  theorem  is  true  when  the  exponent  has  any 
particular  value  (say  k),  then  it  is  also  true  ivhen  the  exponent  has 
the  next  higher  value* 

But,  by  actual  multiplication,  the  theorem  is  known  to  be  true 
when  n  =  2,  hence,  by  what  has  just  been  proved,  it  is  true  when 
71  =  3 ;  again,  since  it  is  now  known  to  be  true  when  n  =  3,  there- 
fore it  is  true  when  w  =  4 ;  *  and  so  on  without  end :  hence  the 
theorem  is  true  for  every  positive  integral  exponent,*  which  was 
to  be  proved. 

EXERCISES 

1.  In  the  expansion  of  (x  +  yy*  what  is  the  exponent  of  y  in  the  2d 
term  ?  in  the  3d  term  ?  in  the  4th  term  ?  in  the  12th  term  ?  in  the  rth 
term?     What  is  the  sum  of  the  exponents  of  x  and  y  in  each  term  ? 

2.  In  the  expansion  of  (x  +  ?/)"  what  is  the  highest  factor  in  the 
denominator  of  the  3d  term  ?  of  the  4th  term  ?  of  the  10th  term  ?  of  the 
rth  term?  How  does  this  factor  compare  with  the  exponent  of  y  in  any 
given  term  ? 

3.  What  is  subtracted  from  n  in  the  last  factor  of  the  numerator^  in 
the  3d  term  of  the  expansion  of  (x  +  yYl  in  the  4th  term?  in  the  5th 
term?  in  the  9th  term?  in  the  rth  term? 

4.  Based  upon  your  answers  to  Exs.  1-3,  write  down  the  6th  term  of 
(x  +  2/)".     Also  write  the  10th  term  ;  the  17th  term ;  and  the  rth  term. 

203.  Binomial  theorem  continued.  Strictly  speaking,  all  that 
was  really  proved  in  §  202  is  that,  for  every  positive  integral 
value  of  the  exponent,  the  first  four  terms  of  the  expansion  follow 
the  law  expressed  by  Eq.  (1) ;  that  all  the  terms  follow  this  law 
will  now  be  shown. 

In  multiplying  Eq.  (2)  of  §  202  by  x  +  y  the  2d  term  of  the 
product  (3)  is  x  times  the  2d  term  plus  y  times  the  1st  term  of 
(2) ;  so,  too,  the  10th  term  of  (3)  would  be  found  by  adding  x 
times  the  10th  term  to  y  times  the  9th  term  of  (2),  and  the  rth 

*  Only  the  first  four  terms  are  given  in  Eqs.  (2)  and  (3) ;  see  §  203  for  com- 
plete proof. 


348  ELEMENTARY  ALGEBRA  [Ch.  XVIII 

term  of  (3)  by  adding  x  times  the  rth  term  to  y  times  the  (r— l)th 
term  of  (2). 

But  the  (r  —  l)th  and  the  rth  terms  of  (2)  are,  respectively, 

1.2-3- .••(r-2)  ^ 

and        A;(A; -  l)(k  -  2)  .•>  (fc  -  r  +  3)(fe  -  r  +  2).^^.,_, 
^"""^  1.2.3....(r-2)(r-l)  ^      "V     ,        ■ 

therefore  the  rth  term  of  (3)  is 

fe(fc-l)(fe-2)..-(fe-r  +  3) 
1.2.3.  •..  (r- 2) 


I 


fc(A:-l)(fe-2)...(A;-r  +  3)(fc-r  +  2)  \  ^-.+2^.-1 
■^  1.2.3....(r-2)(r-l)  j  ^     ' 

(fe+l)fe(A;-l)  ...  (fe-r  +  3)   ._,^,     , 
*  *'  1.2.3.  ...(r-1)  ^     ' 

which  conforms  to  the  law  for  the  rth  term  expressed  by  (1) 
of  §  202.  Hence  the  rth  term  (i.e.,  every  term)  in  (3)  conforms 
to  the  law  expressed  by  (1),  which  was  to  be  proved. 


EXERCISES 

1.  Write  down  the  expansion  of  (a  +  by-,  also  of  {p  —  qY-  Explain 
why  the  alternate  terms  in  the  expansion  of  (p  —  qY  are  negative. 

2.  Write  down  the  first  3  terms  of  {x  +  y)^^ ;  also  the  8th  term. 

3.  Write  down  the  4th  and  7th  terms  of  (a  —  xy^. 

4.  How  many  terms  are  there  in  the  expansion  of  {x  -f  yY^l  Write 
down  the  first  three,  and  also  the  last  three  terms  of  this  expansion,  and 
compare  their  coefficients. 

5.  Write  down  the  coefficient  of  the  term  containing  a^y^  in  (a  —  yy^. 

6.  Expand  (3  a2  _  2  xy^y\  compare  Ex.  2,  p.  93. 

7.  Write  down  the  4th  term  of  (f  a:  -  |  yy^ ;  also  the  9th  term. 

8.  How  many  terms  are  there  in    ( a:  —  ]     ?    Write  down  the  10th 

term.     Also  write  the  5th  term  of  (y^-  +'V~)*" 

9.  Write  down  the  term  of  (3  x*  -  2  a;2)7,  u.,  of  {x'^y  {^  x^  -  2y , 
which  contains  x^. 


203-204]  BINOMIAL    THEOREM  Md 

10.  Write  down  the  term  of  (  a^  — ^  )   which  contains  a^^. 

11.  Expand  (a^ -{■  ^  a^x-^y,  and  write  the  result  with  positive  ex- 
ponents. 

12.  Expand  (1  —  a:  +  x^y  by  means  of  the  binomial  theorem  (cf. 
Ex.  25,  p.  205). 

13.  By  applying  the  law  expressed  in  Eq.  (1)  of  §  202,  show  that  the 
coefficient  of  the  (n  +  l)th  term  of  {x  +  iy)«  is  1 ;  also  show  that  the 
coefficient  of  every  term  thereafter  contains  a  zero  factor,  and  hence  that 
(x  +  yy  contains  only  n  +  1  terms. 

14.  Since  (a  +  !))"■  — {h  +  a)",  show  that  the  coefficients  equally  dis- 
tant from  the  ends  of  (a  +  &)"  are  equal ;  show  this  also  by  comparing 
the  coefficient  of  the  rth  term  from  the  beginning  with  that  of  the  rth 
term  from  the  end  [i.e.,  with  the  (n  —  r  +  2)th  term  from  the  beginning]. 

15.  Show  that  the  sum  of  the  binomial  coefficients,  i.e.,  of  1,  n, 
n(n-\)     n(n-l)(n-2)  •     «„ 

2        '  1-2.3        '••''"  ^  • 

Suggestion.    Let  x  =  ?/  =  1,  after  expanding  {x  +  y)^. 

16.  Show  that  the  sum  of  the  even  coefficients  {i.e.,  the  2d,  4th,  •••)  in 
Ex.  15  equals  the  sum  of  the  odd  coefficients,  and  that  each  sum  is  2**-^ 

Suggestion.    Let  x  =  1  and  y  =—lin  {x  +  yy. 

17.  Show  that  the  coefficient  of  the  rth  term  in  {x  -f  yy  may  be  ob- 
tained by  multiplying  that  of  the  (r  —  l)th  term  by  ^~  ^"^    ,  and  thus 

r  —  1 

show  that  the  binomial  coefficients  increase  numerically  in  going  from 
term  to  term  toward  the  center  (cf.  also  Ex.  14). 

18.  Show  that  the  coefficient  of  the  rth  term  is  numerically  greater 
than  that  of  the  (r  —  l)th  term  so  long  as  r<  \(n  +  3)  ;  and  thus  write 
down  the  term  whose  coefficient  is  greatest  in  the  expansion  of  {x  +  ?/)  ^M 
and  also  in  {x  +  yY^. 

204.  Binomial  theorem  extended.  It  may  be  remarked  in  passing 
that  the  binomial  theorem  (§  202),  which  has  thus  far  been  re- 
stricted to  the  case  where  the  exponent  is  a  positive  integer,  is 
greatly  extended  in  Higher  Algebra,  where  it  is  shown  that,  under 
certain  restrictions,  it  admits  negative  and  fractional  exponents 
also.  Although  the  proof  of  this  fact  is  beyond  the  limits  of  this 
book,  its  correctness  may  be  assumed  in  the  following  exercises. 


350  ELEMENTARY  ALGEBRA  [Ch,  XVIII 

EXERCISES 

1.  By  means  of  the  binomial  theorem  wi^te  the  first  four  terms  of 
(1  +  a:)^ ;  the  first  five  terms  of  (a  +  b)-^;  the  5th  term  of  (1  -  3  a;)i 

2.  Show  that  the  application  of  the  binomial  theorem  to  such  cases  as 
the  above  gives  rise  to  an  infinite  series  (cf.  Ex.  13,  §  203). 

3.  Expand  (1  —  x)-^  to  8  terms  by  the  binomial  theorem  and  compare 
the  result  with  the  first  8  terms  of  the  quotient  1  -f-  (1  —  x). 

4.  Show  that  (25  +  1)^  =  5  + ^V—  nks  +  z<yhws  —  •-■>  when  expanded 
by  the  binomial  theorem  and  simplified ;  compare  this  result  with  V'26 
as  found  by  the  usual  method. 

5.  By  means  of  the  expansion  of  (9  —  2)*  show  how  to  get  an 
approximate  value  of  the  square  root  of  7. 

205.  The  square  of  a  polynomial.  In  §  61  it  was  pointed  out 
that,  by  actual  multiplication,  the  square  of  a  polynomial  consist- 
ing of  3,  4,  or  5  terms,  equals  the  sum  of  the  squares  of  all  the 
terms  of  the  polynomial,  plus  twice  the  product  of  each  term  by 
all  those  that  follow  it.  It  will  now  be  shown  that  if  this  theorem 
is  true  for  polynomials  of  n  terms,  then  it  is  also  true  for  those  of 
n  +  1  terms,  and  from  this  it  will  follow,  as  in  §  201,  that  it  is 
true  for  polynomials  of  any  finite  number  of  terms  whatever,  since 
it  is  already  known  to  be  true  for  polynomials  of  five  terms. 

Let  a-\-h-^c-\ \-p  +  qhe  a.  polynomial  of  n  terms,  and  let 

(a  +  b  +  c-\ [- p -{- qf  =  a^  +  b^ -\ ^q'~-\-2ab  +  2ac-\ \-2aq 

+  2bG-\ \-^bq-{ \-2pq. 

In  this  identity  replace  a  everywhere  hj  x-\-y;  then  the  number 
of  terms  in  the  polynomial  in  the  first  member  will  become  n  4- 1, 
and  the  second  member  will  still  consist  of  the  sum  of  the  squares 
of  all  the  terms  of  the  polynomial,  plus  twice  the  product  of  each 
term  by  all  those  that  follow  it  (the  student  should  work  this  out 
in  detail) ;  therefore,  if  the  theorem  is  true  for  polynomials  of  n 
terms,  then  it  is  also  true  for  those  of  n  +  1  terms,  which  was  to 
be  proved. 

EXERCISES 

Expand : 

1.  (a  +  b-3x  +  2ah-  1)2.  «     /     .  2      „       .2 

2.  (2-3a?  +  4ma:2-3ma;+3a;- 3a2a:)2. 


\        X  ml 


,  APPENDIX  A 

IRRATIONAL  NUMBERS 

[Supplementary  to  §  132] 

206.  Irrational  numbers  are  defined  and  illustrated  in  Chapter  XIV, 
and  it  is  there  tentatively  assumed,  not  only  that  the  earlier  definitions 
of  sum,  product,  etc.,  apply  to  these  numbers,  but  also  that  they  are 
subject  to  the  combinatory  laws  previously  established  for  rational 
numbers. 

These  definitions  will  now  be  restated  from  a  somewhat  broader  point 
of  view,  and  one  from  which  the  proofs  of  the  combinatory  laws  are 
easily  established. 

As  in  §  130,  note  2,  two  infinite  series  may  be  found  such  that  the 
square  of  each  term  of  the  first  series  is  less  than  2,  while  the  square  of 
each  term  of  the  second  series  is  greater  than  2.  These  series  may  be 
conveniently  written  in  the  form 

1,  1.4,  1.41,  1.414,  1.4142,  .••  <V2<2,  1.5,  1.42,  1.415,  .-.;     (1) 

and  the  value  of  V5  may  be  thought  of  as  defined  by  them. 

For,  let  a  point  P  move  along  a  straight  line  AB  va.  such  a  way  that, 
at  successive  stages,  its  distances  from  0  are:  1,  1.4,  1.41,  •••  (shown 
in  the  figure  by  OP^,  OP^,  •••),  and  let  another  point  Q  move  along 

A q T,        p,Q,         g, B 

this  line  so  that  its  distances  from  0  are  successively:  2,  1.5,  1.42,  ... 
(shown  in  the  figure  by  OQj,  OQg^  •-)•  Then  clearly  the  point  P  will 
always  be  at  the  left  of  Q,  —  since  each  number  of  the  first  series  is 
smaller  than  each  number  of  the  second,  —  and  P  and  Q  will  approach 
each  other  infinitely  closely,  but  will  never  meet,  —  since  the  distance 

between  them  at  the  nth  stage  of  their  progress  is  — ,  which  may  be 

made  smaller  than  any  assigned  distance,  however  small,  by  making  n 
sufficiently  large,  but  which  can  not  be  made  zero.  In  other  words :  the 
points  P  and  Q  are  each  approaching,  infinitely  closely,  a  fixed  common  point 
R  which  lies  between  them. 

361 


352  ELEMENTARY  ALGEBRA  [App.  A 

Moreover,  there  exists  only  one  such  fixed  point,  as  R,  between  P  and  Q : 
for,  if  there  be  more  than  one,  let  R^  be  another  point  distinct  from  R, 
and  approached  infinitely  closely  by  both  P  and  Q,  and  let  d  be  the  dis- 
tance between  R  and  R^ ;  now  the  distance  between  P  and  Q  is  — -,  and 

this  may  be  made  smaller  than  d  by  suflBciently  increasing  n ;  therefore 
R  and  R^  can  not  both  be  between  P  and  Q,  which  was  to  be  shown. 

Now,  there  being  one,  and  only  one,  fixed  point,  R,  determined  (de- 
fined) by  the  two  infinite  series  in  (1)  above,  therefore  the  distance  OR 
may  be  said  to  be  defined  by  these  infinite  series;  and  since  these  series 
are  formed  as  above  explained,  therefore  the  distance  OR  may  be 
appropriately  represented  by  the  symbol  V2 ;  hence  the  above  series 
may  be  said  to  de^ne  the  value  of   V2  (cf.  §  130,  note  3). 

As  in  the  particular  example  just  now  considered,  so  in  general,  ani/ 
two  infinite  series  of  rational  numbers  {expressed  decimally 
or  otherwise),  one  series  increasing  and  the  other  decreas- 
ing, define  an  irrational  number  if  the  difference  between 
the  nth  terms  of  the  two  series,  while  it  can  never  be  made 
zero,  may  be  made  smaller  than  any  assigned  number, 
however  small,  by  sufficiently  increasing  n.  Moreover,  every 
irrational  number  may  be  represented  in  this  way  (cf.  §  130). 

207.   Equality,  sum,  product,  etc.,  of  irrational  numbers.     Let  k  and  Id 

be  two  given  positive  irrational  numbers,  and  let  them  be  defined  by 
infinite  series  of  rational  numbers  as  explained  in  §  206 ; 

i.e.,  let  Op  a^,  a^  —  a„,  •••  <k<h^,  h^,  h^,  —  bn,  — ,  (1) 

and  a\,  a'^,  a'g,  ...  a'„,  ...  <k'  <  h\,  h\,  ft'g,  ...  6'„,  ...,  (2) 

wherein  a„  —  bn  and  a'„  —  b'n  may  each  be  made  smaller  than  any 
assigned  number,  however  small,  by  sufficiently  increasing  n. 

Then  k  is  said  to  be  equal  to  k'  if,  and  only  if,  every  one  of  the  a's  is 
less  than  every  one  of  the  6"s,  and  every  a'  is  less  than  every  b. 

And  k  is  said  to  be  greater  than  k'  if,  and  only  if,  some  of  the  a's  are 
greater  than  some  of  the  6"s. 

Again,  the  sum,  difference,  product,  and  quotient  of  k  and  k'  may  be 
defined,  respectively,  by  the  following  pairs  of  infinite  series : 

fli  +  a'l,  flg  +  «'2'  «3  +  «'3»  •••««  +  a'n,  "'  <k  +  k'  <,b^-\-  b\, 

b^  +  b\,  &3  +  fe'a,  -  bn  +  ft'„,  -,      (3) 

a,  -  h\,  a^  -  V^  as  -  b'g,  ...  a„  -  ft'„,  ...  <  ^  -  ^^'  <  &,  -  a'„ 

^2  -  «'■«  h  -  «'3'  --K-  «'«'  — »     (4) 


206-209]  IRBATIONAL  NUMBERS  353 

a^  •  a'l,  Og  •  a'j,  a^  •  a'g,  •••  a»  •  «'„,  •••  <  ^'  •  ^'  <  6j  •  6'^, 

*2  •  *'2>    *3  •  *'3'    •••  ^H  •  *'n,    — ,  (5) 

and    flj  -4-  h\,  a^  h-  6'2»  as  "^  ^V  •-  «n  ^  6'«,  -"  <k-^k'  <\^  a\, 

h^  ^  a\,  63  -  a'g,  ...  />„  -  a'„,  .-.     (6) 

Note  1.  Observe  that  if  k  =  k' ,  as  defined  above,  then  these  two  irrational 
numbers  have  the  same  decimal  expressions,  however  far  they  may  be  carried 
out.  For  suppose  that  some  decimal  figure,  say  the  14th,  in  k  is  greater  than 
the  corresponding  figure  in  k',  then  the  corresponding* a  woulci  be  equal  to,  or 
greater  than,  the  corresponding  6',  and  k  would  not  equSil  k'  under  t^e  above 
definition.  .  '  '  - 

Note  2.  In  applying  the  above  definitions,  say  that  of  the  sum,  it  may  happen 
that  ai  +  a'l  =  03  +  a'2  =  —  =  an  +  a'n  =  —  =  &n  +  b'n  =  •.• ;  in  this  case  k  +  k' 
=  an-{-a'n=  bn  +  b'n,  i.e.,  this  sum  is  a,  rational  number.  To  illustrate  this  fact 
numerically,  let  k  =  y/2  and  k'  —  5~ \/2. 

Note  3.  The  above  definitions  [inequalities  (3)-(6)]  apply  also  when  negative 
irrational  numbers  are  involved :  those  of  sum  and  difference  apply  directly,  and 
those  of  product  and  quotient  apply  by  regarding  the  numbers  as  positive  and 
attaching  the  proper  sign  to  the  result. 

208.  Comparisons  and  operations  between  rational  and  irrational  num- 
bers. A  given  rational  number  r  is  said  to  be  less  than  k  (see  §  207)  if, 
and  only  if,  some  of  the  a's  are  greater  than  r,  otherwise  it  is  greater 
than  k. 

The  sum  of  a  rational  and  an  irrational  number,  say  A:  +  r,  is  defined 
by  the  series 

a^  +  r,   a^-\-r,  flg  +  r,    .- a„+r  <  ^+r  <  ft^  +  r,   h^-^r,   63+r,    —  5„  +  r,    ...; 

and  the  difference,  product,  and  quotient  of  a  rational  and  an  irrational 
number  are  defined  in  a  similar  manner. 

209.  Combinatory  laws  of  irrational  numbers.  That  the  irrational 
numbers  are  subject  to  the  same  combinatory  laws  as  are  the  rational 
numbers  follows  easily  from  the  definitions  given  in  §§  207  and  208. 
Thus,  by  (3)  of  §  207, 

aj  +  a'i,  a2+«'2'  «3+«V  •*•  <  k-\-k'  <h^  +  b\,  h^-\-h\,  b^+b'^,  •-,       (1) 

and  a'l  +  aj,  a'g  +  flg'  ^'3  +  03,  ...  <  A;'  +  ^<6'j  +  ii,  ft'2  +  ^2'  ^'s  +  ^s'  "*  5       (2) 

but  since  the  addition  of  rational  numbers  is  commutative,  i.e.,  since 
flj  -f  a'l  =  a'j  +  flj,  etc.,  therefore  the  two  infinite  series  which  define 
k  +  k'  are  exactly  the  same  as  those  which  define  k'  +  k;  but,  by  §  206, 
two  such  series  define  one,  and  only  one,  irrational  number,  therefore 
k  -hk'  =  k'  +  k. 


354  ELEMENTARY  ALGEBRA  [App.  A 

In  the  same  way  it  may  be  shown  that  the  sum  of  any  number  of 
irrational  numbers  is  independent  of  the  order  in  which  the  summands 
are  arranged ;  i.e.,  irrational  numbers  are  subject  to  the  commutative  law  of 
addition. 

That  this  law  holds  also  when  rational  numbers  are  added  to  irra- 
tional numbers,  and  vice  versa,  follows  from  §  208. 

Moreover,  by  means  of  (5)  of  §  207,  and  by  reasoning  altogether  simi- 
lar to  that  which  has  just  now  been  employed,  the  commutative  law  of 
multiplication  may  be  established. 

The  associative  law  of  addition,  and  also  that  of  multiplication,  is 
proved  from  the  commutative  law  in  precisely  the  same  way  as  that 
employed  for  integers  in  §§  51  and  53. 

And  finally,  since  (I  -]-m)n  =  ln  +  mn  for  all  rational  numbers,  there- 
fore, by  reasoning  altogether  similar  to  that  employed  to  prove  the  com- 
mutative law  of  addition  and  of  multiplication,  it  is  easily  proved  that 
{k  +  k') k"  =  k-  k"  +  k'  •  k",  wherein  k,  k',  and  k"  are  any  three  irrational 
numbers  which  are  defined  by  infinite  series  of  rational  numbers  as  in 
§  207 ;  hence,  even  for  irrational  numbers,  multiplication  is  distributive 
with  regard  to  addition. 

Remark.  For  a  more  extended  treatment  of  irrational  numbers  see  Tannery's 
Arithmetique,  Chapter  XII;  or  Weber's  Encyklopadie  der  Elemen tar-Ma the- 
matik,  Chapter  IV. 


APPENDIX  B 

COMPLEX  NUMBERS 

[Supplementary  to  §  146] 

210.  Complex  numbers.  In  the  treatment  of  complex  numbers  given 
in  the  preceding  pages,  considerations  of  simplicity  demanded  that  the 
proofs  of  their  combinatory  laws  be  postponed  ;  accordingly  these  laws 
were  there  tentatively  assumed  to  hold,  —  compare  footnote,  p.  244. 

The  following  definition  of  a  complex  number,  while  it  may  at  first 
sight  seem  somewhat  arbitrary,  is  fully  justified  by  the  beautiful  results 
to  which  it  leads,  and  it  serves  at  the  same  time  to  illustrate  a  means  of 
defining  numbers  which  has  not  hitherto  been  employed  in  this  book. 

A  complex  number  is  a  combination  of  two  real  numbers,  such  as  a  and 
b,  which  will  be  temporarily  represented  by  the  symbol  (a,  b),  and  which 
satisfies  the  following  defining  equations : 

(a,  h)  =  (a',  h'),  if,  and  only  if,  a  =  a'  and  b  =  b',  (1) 

(a,  b)  +  (a',  b')  =  (a  +  a',  ft  +  b'),  (2) 

and  (a,  b)  -  (a',  b')  =  (aa'  -  bb',  ab'  +  a'b) ;  (3) 

these  equations  merely  define  what  is  meant  by  equals,  sum,  and  product, 
for  complex  numbers. 

Moreover,  in  order  immediately  to  connect  complex  numbers  more 
closely  with  real  numbers,  and  to  make  the  latter  a  special  case  of  the 

^orm^^^\^^  (a,0)=a,  (4) 

which  may  be  done  since  it  is  consistent  with  each  of  the  above  defining 
equations. 

211.  Immediate  consequences  of  the  definitions  in  §  210.  It  will  now 
be  shown  that  if  (a,  b)  is  any  combination  whatever  of  two  real  num- 
bers which  satisfies  the  defining  equations  in  §  210,  then 

(a,  &)  =  a  +  bi, 

wherein  i^  =  —  1 ;  and  hence  that  the  complex  number  defined  in  §  210 
is  none  other  than  the  complex  number  a  +  6i,  already  considered  in 
Chapter  XIV. 

866 


356  ELEMENTARY  ALGEBRA  [App.  B 

Thus,  by  (3)  and  (4)  of  §  210, 

(0,  1).(0,  1)  =  (-1,  0)=-l, 

i.e.,  (0,  1)2  =  -  1,  and  therefore  (0,  1)  =  a^^=^  =  i. 

Again,  by  (3)  of  §  210,       (0,  h)  =  {h,0   •  (0,  1), 

Le.,  (0,  h)  =  hi. 

And  finaUy,  by  (2)  of  §  210, 

(a,&)  =  (a,0)-  ^0,&), 

i.e.,  {a,  b)  =  a  +  hi 

which  was  to  be  proved. 

212.  Combinatory  laws  of  complex  numl  s.  That  the  commutative 
law  of  addition,  already  established  for  reaj  umbers,  holds  for  complex 
numbers  also  may  be  easily  proved  as  follow 

By  (2)  of  §  210, 

(a,  b)  +  (a',  h')  =  (a  +  a',  &  +  V),  and  (a',  h')  +  (a,  h)  =  (a'  +  «,  h'  +  h), 
but,  since  a,  a',  h,  and  h'  are  real  numbers, 
therefore  a  +  a'  —  a'  +  a^  and  b  +  b'  =  b'  +  b, 

and  therefore  (a,  b)  +  (a',  6')  =  (a',  &')  +  (a,  b) ; 

i.e.,  the  commutative  law  holds  for  the  sum  of  two  complex  numbers. 

Moreover,  it  is  evident  that  the  proof  just  now  given  for  two  complex 
numbers  may  be  easily  extended  to  any  number  of  siich  numbers;  and 
since  (a,  b)  is  a  real  number  when  b  =  0,  and  a  pure  imaginary  number 
when  a  =  0,  therefore  this  proof  applies  also  when  real  numbers  and 
complex  numbers  are  added  together. 

Again,  by  means  of  (3)  of  §  210,  and  by  reasoning  altogether  similar 
to  that  which  has  just  been  employed  in  the  proof  of  the  commutative 
law  of  addition,  it  is  easily  shown  that  multiplication  is  also  subject  to 
the  commutative  law. 

The  associative  law  of  addition  and  of  multiplica^ 'on  is  proved  from 
the  commutative  law  in  precisely  the  same  way  u  that  employed  for 
integers,  §§  51  and  53. 

And  finally,  it  is  easily  proved  from  the  definitions  of  §  210  that 

(a,  b)  +  (a',  b')  .  (a",  b")  =  (a,  b)  .  (a",  b")  +  (a',  6;)  •  (a",  b") ;  * 

*  The  details  of  this  proof  are  left  as  an  exercise  for  the  student ;  he  may 
establish  this  equality  by  showing  that  each  member  is  equal  to  the  complex 
number  ^^^„  _  ^^,.  ^  ^.^,.  _  j,^„^  ^^,.  ^  ^„^  ^  ^,^„  ^  ^„^,^ 


211-214]  COMPLEX  NUMBERS  357 

i.e.,  multiplication  with  complex  numbers  is  distributive  with  regard  to 
addition. 

213.  Subtraction  and  divisipn  with  complex  numbers.  Here,  as  with 
real  numbers,  subtraction  and  division  are  defined,  respectively,  as  the 
inverses  of  addition  and  mult  plication  (cf.  §  3)  ;  and,  based  upon  this 
definition,  it  will  now  be  shown  that  any  two  given  complex  numbers 
have  a  unique  difference  and  a  unique  quotient,  which  may  be  easily 
written  down  from  the  giv^en  numbers.  To  show  this,  let  (a,  6)  and 
(a',  h')  be  any  two  given  '    Kiplex  numbers,  and  let 

then,  by  §  3  (ii),  (x        +  (a',  h')  =  (a,  h), 

•6 
whence,  from  (2)  and  (1)  ,,  i§  210, 

X  -\' i'  =  a  and  y  +  b'  =  b; 

therefore  x  =  a  —  a'  and  y  =  h  —  b', 

i.e.,  (a,  b)  -  (a',  b')  =  (a  -  a',  b  —  b'). 

Again,  let  '    (a,  b)  ^  («',  b')  -  {x,  y)  ; 

then,  by  §  3  (iv),       (x,  y)  ■  (a',  b')  =  (a,  6), 

and  therefore,  from  (3)  and  (1)  of  §  210, 

a'x  —  b'y  =  a  and   a'y  +  b'x  =  5, 

1  aa'  +  bb'        -,  a'b  —  ab' 

whence  x  =     ,,^      ,„,    and  y  = 


2.0.. 


(g,  b)  _  lag'  +  bb'    oH)  —  ab'\ 
(aT6^~  U'2  +  6'2'    a'^+b'^l' 


On  recalling  the  conclusion  of  §211,  the  two  results  just  obtained 
may  be  written,  re^s^ectively,  as 

a  -^^i  -  (a'  +  b'i)  =  a  -  a'  +  (6  -  b')i, 

A  g  +  bi  _  aa'  +  hb'  +  ja'h  -  ab')i 

214.  Powers  and  roots  of  complex  numbers.  Raising  a  complex 
number  to  a  positiv^i  integral  power  is  merely  a  special  case  of  multiplica- 
tion, and  is  therefore  fully  provided  for  in  (3)  of  §  210. 


358 


ELEMENTARY  ALGEBRA 


[App.  B 


The  method  of  extracting  the  square  root  of  a  complex  number  *  is 
illustrated  by  means  of  a  particular  example  in  §  182;  and  it  is  evident, 
from  what  is  there  said,  that  this  same  process  may  be  applied  to  any 
complex  number  whatever. 

Moreover,  by  the  method  employed  in  the  note  of  §  182,  it  is  now  evi- 
dent that  any  even  root  of  any  negative  number  whatever  can  be 
expressed  in  the  form  a  +  bi,  wherein  a  and  b  are  real,  and  ^2  =  —  1. 

215.  Graphic  representation  of  complex  numbers.  A  complex  number, 
such  as  a  +  bi,  may  be  graphically  represented  by  the  point  (P) 
whose  coordinates  (§  114)  are  a  and  b.  In  this  scheme  of  representation 
it  is  evident  that  to  every  complex  number  there  corresponds  one  and  only 
one  point  in  the  plane,  and  conversely,  to  every  point  in  the  plane  there 
corresponds  one  and  only  one  complex  number, — if  a=0  the  correspond- 
ing point  lies  on  the  line  OY,  while  if  &  =  0  it  lies  on  OX. 


y 

X        ct 

J 

/ 

b 

o 

0 

Jt 

I 

X 

This  method  of  graphically  representing  a  complex  number  was  intro- 
duced by  Argand  in  1806,  and  is  known  as  the  Argand  diagram. 

Instead  of  representing  a  +  bi  by  the  point  P,  it  may  also  be  repre- 
sented by  the  line  OP ;  each  of  these  methods  is,  in  fact,  often  employed. 

The  length  of  the  line  OP,  which  is  Va^  +  b%  is  called  the  modulus 
(also  the  absolute  value)  of  the  number  a  +  bi,  and  the  angle  XOP  is 
called  its  argument  (also  its  amplitude). 

Not  only  may  given  complex  numbers  be  represented  by  the  Argand 
diagram,  but  the  sum,  product,  etc.,  of  two  or  more  of  them,  being  itself 
a  complex  number,  may  also  be  represented  by  such  a  diagram. 

E.g.,  in  the  following  diagram,  OP  represents  9  +  2{,  OQ  represents 
2  -f  7 1,  and  OR  represents  their  sum,  viz.,  11  +  9  i. 

Observe  that  PR  is  equal  and  parallel  to  OQ  (why?),  and  hence 
that  OPRQ  is  a  parallelogram.     From  this  it  follows  that  if  any  two 


*  Higher  roots  of  complex  numbers  can  not  in  general  be  extracted  by  the 
elementary  methods  thus  far  studied. 


214-216] 


COMPLEX  NUMBERS 


359 


F 

R 

g....,- -y^ 

1 X          1 

j^y^^^^^" 

0 

X 

complex  numbers  are  represented  by  the  Argand  diagram,  then  their  sum 
is  represented  by  the  diagonal  of  the  parallelogram  of  which  the  given 
numbers  are  a  pair  of  adjacent  sides. 

Note  1.  From  a  physical  point  of  view,  it  is  also  quite  appropriate  to  call  OR 
the  swn  of  OF  and  OQ.  Thus,  if  two  forces  which  are  represented  in  amount 
and  direction  by  OP  and  OQ,  respectively,  act  simultaneously  upon  a  body 
situated  at  0,  the  result  is  the  same  as  if  a  single  force  represented  in  amount 
and  direction  by  OR  were  acting  on  this  body. 

Note  2.  The  fact  that  t  •  i  =—  1  is  also  consistent  with  the  Argand  diagram. 
E.q.,  the  effect  of  multiplying  any  given  line  as  OM  by  —  1  is  to  reverse  its 
quality,  and  this  may  be  thought  of  as  accomplished  by  rotating  OM  through 
an  angle  of  180°  to  the  position  OM' ,  as  shown  in  the  figure;  now,  since  multi- 


H 


M' 


O 


plying  OM  hy  i-i  also  reverses  its  quality,  therefore  multiplying  OM  by  i  alone 
should  rotate  it  through  90°  to  the  position  OH.  Hence  if  OM  represents  any- 
real  number,  then  the  number  represented  by  i  •  OM  should  be  laid  off  on  a  line 
perpendicular  to  OM,  as  it  is  in  the  Argand  diagram. 


INDEX 


[Numbers  refer  to  pages.] 


Absolute,  term,  267. 

value,  21,  358. 
Addition,  2,  11,  44,  45,  131. 

of  negative  numbers,  23. 
Algebraic,  expressions,  30,  31,  58. 

numbers,  21,  26,  28. 

sum,  25. 
Amplitude,  358. 
Antecedent,  318,  320. 
Argand  diagram,  358. 
Argument  of  complex  numbers,  358. 
Arithmetical,  means,  325. 

numbers,  21. 

operations,  order  of,  13. 

processes,  2." 

progression,  331. 
Arithmetico-geometric  series,  342. 
Arrangement  of  expressions,  58. 
Associative  law,  52,  76,  79. 
Axioms,  33. 

Base  of  power,  12,  201. 
Binomial,  42. 

square  of,  87. 

theorem,  92,  346. 
Brace,  bracket,  etc.,  14. 

Character  of  roots,  277. 
Checks,  56  (Ex.  7),  57,  67. 
Clearing  equations  of  fractions,  35. 
Coefficients,  42. 

detached,  61,  71. 

numerical  and  literal,  42. 
Commensurable  numbers,  319. 
Common,  difference,  331. 

factors,  112. 

ratio,  336. 


Commutative  law,  52,  74,  77. 
Completing  the  square,  269. 
Complex,  factors,  251. 

fractions,  83,  137. 

complex  numbers,  244,  355. 

graphic  representation  of,  358. 

square  roots  of,  311. 
Conditional  equation,  32. 
Conjugate  surds  and  imaginaries,  242, 

248. 
Consequent,  318,  320. 
Constants,  327. 
Continuation  symbols,  4. 
Continued  product,  27. 
Continued  proportion,  321. 
Cube  roots,  216,  219. 

of  unity,  286  (Ex.  23). 

Decreasing  series,  339. 
Degree,  of  terms,  etc.,  59. 

of  an  equation,  141. 
Detached  coefficients,  61,  71. 
Difference,  3. 
Discriminant,  277. 
Distributive  law,  55. 
Division,  4,  13,  28,  64,  66. 
Divisor,  dividend,  4. 

Elimination,  167,  169,  170,  297. 
Equations,  2,  32,  33,  35. 

consistent,  simultaneous,  etc.,  162, 

165,  177. 
equivalent,  143. 
fractional,    irrational,    etc.,    147, 

282,  283. 
graphic  representation,  189,  314. 
graphic  solution,  316,  316, 


361 


362 


INDEX 


Equations,  indeterminate,  162. 

in  quadratic  form,  291. 

irrational  and  radical,  283. 

literal,  141,  145,  177. 

numerical,  141. 

of  a  problem,  37. 

quadratic,  267,  298,  306,  314. 

reciprocal,  293. 

simple,  linear,  etc.,  142. 

solution  of,  33,  109,  165,  303. 

symmetric,  304. 
Equivalent  equations,  143. 
Evolution,  205. 
Expanded  products,  60. 
Exponents,  12,  63,  252. 

laws  of,  53,  62,  201,  259. 
Expressions,  arrangement  of,  30,  58. 
Extremes  and  means,  320,  335,  341, 
343. 

Factor  theorem,  100. 

Factoring,  solving  equations  by,  94, 

96,  109,  274. 
Factors,  H.  C.  F. ,  and  complex,   94, 

112,  251. 
Formulas,  for  A.  P.  and  G.  P.,  333, 
337. 
for  solving  equations,  16, 145,  178, 
276. 
Fourth  proportional,  320. 
Fractional,  equations,  147,  282. 
exponents,  252,  261. 
powers,  253. 
Fractions,  13,  80,  83,  126,  137. 

Geometric  progression,  etc.,  336,  341. 

infinite  G.  P.,  339. 
Graphic,  representation  and  solution 
of   equations,    189,    192,    314, 
315. 
representation  of  complex  num- 
bers, 368. 

Harmonic  series,  342. 
Higher  roots,  221. 
Highest  common  factor,  112. 
Homogeneous    equations,    etc.,    59, 
141. 


Identical  equations,  identity,  32. 
Imaginary  numbers,   224,   244,   245, 

250,  311. 
Increasing  series,  339. 
Incommensurable  numbers,  319. 
Incompatible  equations,  165. 
Inconsistent  equations,  165. 
Independent  equations,  165. 
Indeterminate  equations,  162. 

system,  187. 
Induction,  mathematical,  344. 
Inequalities,  193,  194,  197. 
Infinite  and  finite  numbers,  86. 

series,  331. 
Infinite  G.  P.,  339. 
Inserting  parentheses,  50. 
Integral,  equation,  141. 

expressions,  41. 
Interpretation  of  solutions,  167. 
Inverse  operations,  2. 
Involution,  201. 

Irrational  numbers  and  equations,  224, 
283,  351. 

Known  and  unknown  numbers,  141. 

Law,  of  exponents,  53,  62,  201,  259. 

of  signs  in  multiplication  and  divi- 
sion, 26,  29. 
Letter  of  arrangement,  59. 
Like  and  unlike  terms,  43. 
Literal  equations,  141,  145,  177. 
Literal  notation,  5,  7,  15. 

advantages  of,  7,  15. 
Lowest  common  multiple,  122. 

Mathematical  induction,  344. 
Maximum  and  minimum  values,  294. 
Mean  proportional,  320. 
Means,    extremes,    etc.,     336,     341, 

343. 
Minuend,  3. 
Modulus,  358. 
Monomials,    addition,    etc.,   42,    44, 

46. 
Multiples,  L.  C.  M.,  etc.,  122. 
Multiplicand,  multiplier,  3. 
Multiplication,  etc.,  3,  12,  62,  69. 


INDEX 


363 


Negative,  exponents,  63. 

numbers,  18-21,  23,  24. 

terms,  31,  43. 
Numbers,  absolute  value  of,  21. 

commensurable,  etc.,  319. 

constants  and  variables,  327. 

finite  and  infinite,  86. 

imaginary  and  complex,  224,  244, 
311,  355. 

known  and  unknown,  141. 

literal,  5. 

natural,  positive,  etc.,  1,  18,  20,  21. 

opposite,  21. 

prime  and  composite,  94. 

rational  and  irrational,  224,  351. 

real,  224. 

Operations  with  literal  numbers,  11. 
Opposite  numbers,  21. 
Order  of  operations,  13. 

Parentheses,  14,  49,  50. 
Polynomials,   addition,   etc.,  42,    44, 
48. 

square  of,  91,  350. 
Positive  numbers,  terms,  etc.,  20,  31, 

43. 
Prime  factors,  94. 

unique  set  of,  122. 
Principal  roots,  227. 
Principles  of  clearing  of  fractions,  149. 

of  elimination,  170,  298,  306. 

of  H.  C.  F.,  119. 

of  inequalities,  194. 

of  proportion,  321. 
Problems,  directions  for  solving,  36. 

equations  of,  37. 

general,  157. 
Products,  3,  26,  53,  55,  57,  etc. 

of  sum  and  difference,  89. 
Progression,  arithmetical,  331. 

geometric,  336. 

harmonic,  342. 
Proof  by  induction,  344. 
Property,  of  complex  numbers,  250. 

of  quadratic  surds,  243. 
Proportion,  its  principles,  320,  321. 

abbreviated,  328. 


Quadratic  equations,  267. 

graphs  of,  314. 

principles  involved,  298,  306. 

special  devices  for,  303. 
Quadratic  surds,  property  of,  243. 

Radicals,  radical  equations,  226,  383. 
Radicand,  206. 
Ratio,  318,  319. 

common,  in  G.  P.,  336. 
Rational  numbers,  224, 
Rationalizing  factor,  242,  264. 
Real  numbers,  224. 
Recapitulation,  17,  31,  225  (note  5). 
Reciprocal,  equations,  293. 

of  a  number,  83. 
Remainder,  3. 

theorem,  71,  100. 
Removal  of  parentheses,  49. 
Roots  of  an  equation,  33. 

character  of  roots,  277. 

sum  and  product  of,  280. 
Rule  of  signs,  26. 

Series,  331,  336,  339,  342. 
Signs,  of  aggregation,  13,  14. 

of  operation,  2,  3,  4. 

of  quality,  21,  29. 

of  relation,  2,  193. 
Similar  and  dissimilar  terms,  43. 
Simple  equations,  142. 

one  and  but  one  solution  for,  145, 
178. 
Simultaneous    equations,    165,     174, 

183,  297. 
Solution,  of  equations,  33,  109,  165. 

graphic  method,  315. 

by  special  devices,  303. 
Specific  gravity,  154. 
Square  of  polynomial,  91,  350. 
Square  roots,  209,  213. 

of  quadratic  surds,  etc.,  310,  311. 
Subtraction,  2,  3,  11,  24,  46,  48,  141. 
Subtrahend,  3. 

Sum,  summands,  etc.,  2,  26,  352,  355. 
Surds,  226. 

Symbols  of  continuation,  4. 
Symmetric  equations,  304. 


364 


System  of  equations,  166. 
indeterminate,  187. 


INDEX 

Type  forms,  87. 
Unknown  numbers,  141. 


Term,  absolute,  267. 

Terms,  positive,  negative,  etc.,  30,  31, 

43. 
Theorem,  binomial,  92,  344. 
Third  proportional,  321. 
Transposition,  36. 
Trinomial,  42. 


Variables,  variation,  327. 
Verification,  33. 
Vinculum,  14. 

Zero,  exponent,  63. 
operations  with,  84. 


V     ,       or 


MODERN    MATHEMATICAL    SERIES 

Elementary  Plane  Geometry 

By  JAMES  McMAHON 

Assistant  Professor  of  Mathematics  in  Cornell  University 

Price,  90  Cents 

PLAN  OF  THE  BOOK 

A  combination  of  demonstrative  and  inventional  geometry. 
The  subject  is  presented  with  Euclidean  rigor;  but  this  rigor 
consists  more  in  soundness  of  structural  development  than  in 
great  formality  of  expression. 

METHOD  OF  ARRANGEMENT 

The  general  enunciation  is  placed  first  and  printed  in  italics. 
Next  comes  the  special  arrangement,  consisting  of  the  special 
statement  of  the  hypothesis,  followed  by  the  diagram  and 
the  special  statement  of  the  conclusion  immediately  following 
the  diagram.  The  successive  steps  in  the  demonstration  lead- 
ing from  hypothesis  to  conclusion  are  then  made  clear  with 
reference  to  the  figure,  the  previous  authority  for  each  step 
being  quoted  or  referred  to. 

SPECIAL  FEATURES 

1.  Theorems  and  problems  are  arranged  in  natural  groups 
with  reference  to  their  underlying  principles. 

2.  Elementary  ideas  of  logic  are  introduced  from  the  begin- 
ning, and  their  significance  for  geometry  is  clearly  shown. 

3.  Typical  forms  of  theorems,  etc.,  are  given  before  the 
special  forms  are  developed. 

4.  Independence  of  reasoning  is  fostered  by  compelling  the 
student  to  rely  on  the  propositions  already  proved. 

5.  Ordinary  size-relations  are  treated  in  a  geometrical  man- 
ner. Words  suggestive  of  length,  area,  distance,  etc.,  are 
referred  to  only  in  Book  VI. 

6.  Instead  of  the  numerical  theory  of  ratio  and  proportion 
usually  given,  the  Euclidean  doctrine  of  ratio  and  proportion  is 
presented  in  a  modernized  form,  emphasizing  its  naturalness 
and  generality. 

7.  The  work  throughout  aims  to  develop  the  student's 
powers  of  invention  and  generalization. 


AMERICAN    BOOK    COMPANY 

t7»J 


THE     MODERN    (Cornell) 
MATHEMATICAL    SERIES 

LUCIEN  AUGUSTUS  WAIT 

General  Editor 
Senior  Professor  of  Mathematics  In  Cornell  University 


ANALYTIC  GEOMETRY 

By  J.  H.  Tanner,  Ph.D.,  Assistant  Professor  of  Mathe- 
matics, Cornell  University,  and  Joseph  Allen,  A.M.,  Instruc- 
tor in  Mathematics  in  the  College  of  the  City  of  New 
York.     Cloth,  8vo,  410  pages $2.00 

DIFFERENTIAL  CALCULUS 

By  James  McMahon,  A.M.,  Assistant  Professor  of  Mathe- 
matics, Cornell  University,  and  Virgil  Snyder,  Ph.D., 
Instructor  in  Mathematics,  Cornell  University.  Cloth,  8vo, 
351  pages $2.00 

INTEGRAL  CALCULUS 

By  Daniel  Alexander  Murray,  Ph.D.,  Instructor  in  Mathe- 
matics in  Cornell  University.     Cloth,  8vo,  302  pages,  $2.00 

DIFFERENTIAL  AND  INTEGRAL  CALCULUS 

By  Virgil  Snyder,  Ph.D.,  Instructor  in  Mathematics,  Cornell 
University,  and  John  Irwin  Hutchinson,  Ph.D.,  Instructor 
in  Mathematics,  Cornell  University.  Cloth,  Svo,  320 
pages $2.00 

ELEMENTARY  GEOMETRY-PLANE 

By  James  McMahon,  Assistant  Professor  of  Mathematics 
in  Cornell  University.    Half  leather,  12mo,  358  pages,  $0.90 

ELEMENTARY  ALGEBRA 

By  J.  H.  Tanner,  Ph.D.,  Assistant  Professor  of  Mathematics, 
Cornell  University.     Half  leather,  Svo,  374  pages      .     $1.00 


The  advanced  books  of  this  series  treat  their  subjects  in  a 
way  that  is  simple  and  practical,  yet  thoroughly  rigorous  and 
attractive  to  both  teacher  and  student.  They  meet  the  needs 
of  students  pursuing  courses  in  engineering  and  architecture  in 
any  college  or  university.  Since  their  publication,  they  have 
received  the  general  and  hearty  approval  of  teachers,  and  have 
been  very  widely  adopted. 

The  elementary  books  will  be  designed  to  implant  the  spirit 
of  the  other  books  into  secondary  schools,  and  will  make  the 
work  in  mathematics,  from  the  very  start,  continuous  and  har- 
monious.   

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Lessons  in  Physical  Geography 

By  CHARLES  R.  DRYER,  M.A.,  F.G.S.A. 

Professor  of  Geography  in  the  Indiana  State  Normal  School 


Half  leather,  12mo.     Illustrated.     430  pages,      .       .       ,       Price,  $1.20 


EASY  AS  WELL  AS  FULL  AND  ACCURATE 

One  of  the  chief  merits  of  this  text-book  is  that  it  is  simpler  than 
any  other  complete  and  accurate  treatise  on  the  subject  now  before  the 
public.  The  treatment,  although  specially  adapted  for  the  high  school 
course,  Is  easily  within  the  comprehension  of  pupils  in  the  upper  grade 
of  the  grammar  school. 

TREATMENT  BY  TYPE  FORMS 

The  physical  features  of  the  earth  are  grouped  according  to  their 
causal  relations  and  their  functions.  The  characteristics  of  each  group 
are  presented  by  means  of  a  typical  example  which  is  described  in  unusual 
detail,  so  that  the  pupil  has  a  relatively  minute  knowledge  of  the  type  form. 

INDUCTIVE  GENERALIZATIONS 

Only  after  the  detailed  discussion  of  a  type  form  has  given  the  pupil 
a  clear  and  vivid  concept  of  that  form  are  explanations  and  general  prin- 
ciples introduced.  Generalizations  developed  thus  inductively  rest  upon 
an  adequate  foundation  in  the  mind  of  the  pupil,  and  hence  cannot 
appear  to  him  mere  formulae  of  words,  as  is  too  often  the  case. 

REALISTIC  EXERCISES 

Throughout  the  book  are  many  realistic  exercises  which  include  both 
field  and  laboratory  work.  In  the  field,  the  student  is  taught  to  observe 
those  physiographic  forces  which  may  be  acting,  even  on  a  small  scale, 
in  his  own  immediate  vicinity.  Appendices  (with  illustrations)  give  full 
instructions  as  to  laboratory  material  and  appliances  for  observation  and 
for  teaching. 

SPECIAL  ATTENTION  TO  SUBJECTS  OF  HUMAN  INTEREST 

While  due  prominence  is  given  to  recent  developments  in  the  study, 
this  does  not  exclude  any  link  in  the  chain  which  connects  the  face  of  the 
earth  with  man.  The  chapters  upon  life  contain  a  fuller  and  more 
adequate  treatment  of  the  controls  exerted  by  geographical  conditions 
upon  plants,  animals,  and  man  than  has  been  given  in  any  other  similar 
book. 

MAPS  AND  ILLUSTRATIONS 

The  book  is  profusely  illustrated  by  more  than  350  maps,  diagrams, 
and  reproductions  of  photographs,  but  illustrations  have  been  used  only 
where  they  afford  real  aid  in  the  elucidation  of  the  text. 


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Gateway  Series  of  English  Texts 

General  Editor,  Henry  van  Dyke,  Princeton  University 

The  English  Texts  which  are  required  for  entrance  to  college, 
edited  by  eminent  authorities,  and  presented  in  a  clear,  helpful, 
and  interesting  form.  A  list  of  the  volumes  and  of  their  editors 
follows.  More  detailed  information,  with  prices  and  terms  for 
introduction,  will  be  gladly  supplied  on  request. 

Shakespeare's  Merchant  of  Venice.  Professor  FeHx  E.  Schelling, 
University  of  Pennsylvania. 

Shakespeare's  Julius  Caesar.  Dr.  Hamilton  W.  Mabie,  "The 
Outlook." 

Shakespeare's  Macbeth.  Professor  T.  M.  Parrott,  Princeton 
University. 

Milton's  Minor  Poems.  Professor  Mary  A.  Jordan,  Smith 
College. 

Addison's  Sir  Roger  de  Coverley  Papers.  Professor  C.  T.  Win- 
chester, Wesleyan  University. 

Goldsmith's  Vicar  of  Wakefield.  Professor  James  A.  Tufts, 
Phillips  Exeter  Academy. 

Burke's  Speech  on  Conciliation.  Professor  William  MacDonald, 
Brown  University. 

Coleridge's  The  Ancient  Mariner.  Professor  George  E.  Wood- 
berry,  Columbia  University. 

Scott's  Ivanhoe.  Professor  Francis  H.  Stoddard,  New  York 
University. 

Scott's  Lady  of  the  Lake.  Professor  R.  M.  Alden,  Leland  Stan- 
ford, Jr.  University. 

Macaulay's  Milton.     Rev.  E.  L.  Gulick,  Lawrenceville  School. 

Macaulay's  Addison.  Professor  Charles  F.  McClumpha,  Uni- 
versity of  Minnesota. 

Carlyle's  Essay  on  Burns.  Professor  Edwin  Mims,  Trinity 
College,  North  Carolina. 

George  Eliot's  Silas  Marner.  Professor  W.  L.  Cross,  Yale 
University. 

Tennyson's  Princess.  Professor  Katharine  Lee  Bates,  Wellesley 
College. 

Scott's  Lady  of  the  Lake.  Professor  R.  M.  Alden,  Leland  Stan- 
ford Jr.  University. 

Tennyson's  Gareth  and  Lynette,  Lancelot  and  Elaine,  and  The 
Passing  of  Arthur.  Dr.  Henry  van  Dyke,  Princeton  Uni- 
versity. 

Irving's  Life  of  Goldsmith. 

Macaulay's  Life  of  Johnson. 


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Text- Books  in  Ancient  History 

BY 

WILLIAM   C.   MOREY,  Ph.D. 

Professor  of  History  and  Political  Science,  University  of  Rochester 


MOREY»S  OUTLINES  OF  ROMAN  HISTORY     .      $1.00 

IN  this  history  the  rise,  progress,  and  decay  of  the  Roman 
Empire  have  been  so  treated  as  to  emphasize  the  unity  and 
continuity  of  the  narrative;  and  the  interrelation  of  the 
various  periods  is  so  clearly  shown  that  the  student  appreciates 
the  logical  and  systematic  arrangement  of  the  work.  The 
scope  of  the  book  covers  the  whole  period  of  Roman  history, 
from  the  foundation  of  the  city  to  the  fall  of  the  Western 
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to  the  exclusion  of  minute  and  unnecessary  details.  The  work 
is  admirably  adapted  to  the  special  kind  of  study  required  by 
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This  is  all  drawn  from  authentic  sources. 

MOREY»S  OUTLINES  OF  GREEK  HISTORY    .     $1.00 

THIS  forms,  with  the  "Outlines  of  Roman  History,"  a 
complete  elementary  course  in  ancient  history.  The 
mechanical  make-up  of  the  volume  is  most  attractive — 
the  type  clear  and  well  spaced,  the  illustrations  well  chosen 
and  helpful,  and  the  maps  numerous  and  not  overcrowded 
with  names.  The  treatment,  therefore,  gives  special  attention 
to  the  development  of  Greek  culture  and  of  political  institu- 
tions. The  topical  method  is  employed,  and  each  chapter  is 
supplemented  by  selections  for  reading  and  a  subject  for  special 
study.  The  book  points  out  clearly  the  most  essential  facts 
in  Greek  history,  and  shows  the  important  influence  which 
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Text-Books  in  Geology 

By  JAMES  D.  DANA,  LL.D. 
Late  Professor  of  Geology  and  Mineralogy  in  Yale  University, 

DANA'S  GEOLOGICAL  STORY  BRIEFLY  TOLD  .  .  .  $1.15 
A  new  and  revised  edition  of  this  popular  text-book  for  beginners  in 
the  study,  and  for  the  general  reader.  The  book  has  been  entirely 
rewritten,  and  improved  by  the  addition  of  many  new  illustrations  and 
interesting  descriptions  of  the  latest  phases  and  discoveries  of  the  science. 
In  contents  and  dress  it  is  an  attractive  volume,  well  suited  for  its  use. 

DANA'S  REVISED  TEXT-BOOK  OF  GEOLOGY  .  .  .  $1.40 
Fifth  Edition,  Revised  and  Enlarged.  Edited  by  William  North 
Rice,  Ph.D.,  LL.D.,  Professor  of  Geology  in  Wesleyan  University. 
This  is  the  standard  text-book  in  geology  for  high  school  and  elementary 
college  work.  While  the  general  and  distinctive  features  of  the  former 
work  have  been  preserved,  the  book  has  been  thoroughly  revised,  enlarged, 
and  improved.  As  now  published,  it  combines  the  results  of  the  life 
experience  and  observation  of  its  distinguished  author  with  the  latest 
discoveries  and  researches  in  the  science. 

DANA'S  MANUAL  OF  GEOLOGY $5.00 

Fourth  Revised  Edition.  This  great  work  is  a  complete  thesaurus  of 
the  principles,  methods,  and  details  of  the  science  of  geology  in  its 
varied  branches,  including  the  formation  and  metamorphism  of  rocks, 
physiography,  orogeny,  and  epeirogeny,  biologic  evolution,  and  paleon- 
tology. It  is  not  only  a  text-book  for  the  college  student  but  a  hand- 
book for  the  professional  geologist.  The  book  was  first  issued  in  1862, 
a  second  edition  was  published  in  1874,  and  a  third  in  1880.  Later 
investigations  and  developments  in  the  science,  especially  in  the  geology 
of  North  America,  led  to  the  last  revision  of  the  work,  which  was  most 
thorough  and  complete.  This  last  revision,  making  the  work  substantially 
a  new  book,  was  performed  almost  exclusively  by  Dr.  Dana  himself,  and 
may  justly  be  regarded  as  the  crowning  work  of  his  life. 


Copies  of  any  of  Dana* s  Geologies  will  be  sent,  prepaid,  to  any  address  on 
receipt  of  the  price. 

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AVERY'S  PHYSICS 

By  ELROY  M.  AVERY,  Ph.D.,  LL.D. 


AVERY'S   SCHOOL    PHYSICS       $1.25 

For  Secondary  Schools 

Avery's  School  Physics  combines  in  one  volume  many 
features  which  are  invaluable  in  a  high  school  course.  Although 
of  great  comprehensiveness,  it  is  concise  and  simple.  It  fur- 
nishes a  text  which  develops  in  logical  order  the  various  divi- 
sions and  subdivisions  of  the  science,  stating  the  fundamental 
principles  with  great  accuracy  and  clearness,  and  consequently 
affording  an  excellent  basis  for  the  student  to  use  in  his  work. 
At  the  same  time  there  are  included  a  large  number  of  exercises 
and  experiments  which  are  amply  sufficient  for  class-room  demon- 
stration and  laboratory  practice. 


AVERY'S   ELEMENTARY   PHYSICS $1.00 

A  Short  Course  for  High  Schools 

This  book  meets  the  wants  of  schools  that  cannot  give  to 
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and  yet  demand  a  book  that  is  scientifically  accurate  and  up- 
to-date  in  every  respect.  While  following  the  general  lines  of 
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effort  and  ability,  it  contains  much  matter  that  is  new  and 
especially  suited  for  more  elementary  work. 


AVERY    AND     SINNOTT'S    FIRST     LESSONS     IN 

PHYSICAL    SCIENCE $0.60 

For  Grammar  Schools 

A  work  adapted  to  the  capacities  of  grammar  school  pupils, 
which  wisely  selects  topics  that  are  fundamental  and  immedi- 
ately helpful  in  other  studies,  as  physical  geography  and  physi- 
ology. The  book  is  of  great  value  to  all  pupils  unable  to  take 
a  high  school  course  in  this  branch.  Although  very  elementary, 
it  is  also  scientifically  accurate.  Step  by  step,  the  pupil  is 
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principles. 


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A  Modern  Chemistry 

Elementary  Chemistry 

$1.10 

LaLborsLtory  MaLnuad 

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By   F.  W.  CLARKE 

Chief  Chemist  of  the  United 
States  Geological  Survey 

and    L.  M.  DENNIS 

Professor  of  Inorganic  and  Analytical 
Chemistry  in  Cornell  University 


THE  study  of  chemistry,  apart  from  its  scientific  and 
detailed  applications,  is  a  training  in  the  interpretation 
of  evidence,  and  herein  lies  one  of  its  chief  merits  as 
an  instrument  of  education.  The  authors  of  this  Elementary 
Chemistry  have  had  this  idea  constantly  in  mind:  theory  and 
practice,  thought  and  application,  are  logically  kept  together, 
and  each  generalization  follows  the  evidence  upon  which  it 
rests.  The  application  of  the  science  to  human  affairs,  and 
its  utility  in  modern  life,  are   given  their  proper  treatment. 

The  Laboratory  Manual  contains  directions  for  experi- 
ments illustrating  all  the  points  taken  up,  and  prepared  with 
reference  to  the  recommendations  of  the  Committee  of  Ten 
and  the  College  Entrance  Examination  Board.  Each  alter- 
nate page  is  left  blank  for  recording  the  details  of  the  experi- 
ment, and  for  writing  answers  to  suggestive  questions  which 
are  introduced  in  connection  with  the  work. 

The  books  reflect  the  combined  knowledge  and  experi- 
ence of  their  distinguished  authors,  and  are  equally  suited 
to  the  needs  both  of  those  students  who  intend  to  take  a 
more  advanced  course  in  chemical  training,  and  of  those 
who  have  no  thought  of  pursuing  the  study  further. 

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Outlines   of  Botany 

FOR   THE 

HIGH  SCHOOL  LABORATORY  AND  CLASSROOM 

BY 

ROBERT   GREENLEAF   LEAVITT,  A.M. 
Of  the  Ames  Botanical  Laboratory 

Prepared  at  the  request  of  the  Botanical  Department  of  Harvard 
University 


LEAVITT'S  OUTLINES  OF  BOTANY.  Cloth,  8vo.  272  pages  .  $1.00 
With  Gray's  Field,  Forest,  and  Garden  Flora,  791  pp.  .  .1.80 
With  Gray's  Manual,  1087  pp 2.25 

This  book  has  been  prepared  to  meet  a  specific  demand.  Many 
schools,  having  outgrown  the  method  of  teaching  botany  hitherto 
prevalent,  find  the  more  recent  text-books  too  difficult  and  comprehensive 
for  practical  use  in  an  elementary  course.  In  order,  therefore,  to  adapt 
this  text-book  to  present  requirements,  the  author  has  combined  with 
great  simplicity  and  definiteness  in  presentation,  a  careful  selection  and 
a  judicious  arrangement  of  matter.     It  offers 

1.  A  series  of  laboratory  exercises  in  the  morphology  and  physiology 

of  phanerogams. 

2.  Directions  for  a  practical  study  of  typical  cryptogams,  represent- 

ing the  chief  groups  from  the  lowest  to  the  highest. 

3.  A  substantial  body  of  information  regarding  the  forms,  activities, 

and  relationships  of  plants,  and  supplementing  the  laboratory 
studies. 

The  laboratory  work  is  adapted  to  any  equipment,  and  the  instruc- 
tions for  it  are  placed  in  divisions  by  themselves,  preceding  the  related 
chapters  of  descriptive  text,  which  follows  in  the  main  the  order  of 
topics  in  Gray's  Lessons  in  Botany.  Special  attention  is  paid  to  the 
ecological  aspects  of  plant  life,  while  at  the  same  time  morphology  and 
physiology  are  fully  treated. 

There  are  384  carefully  drawn  illustrations,  many  of  them  entirely 
new.  The  appendix  contains  full  descriptions  of  the  necessary  laboratory 
materials,  with  directions  for  their  use.  It  also  gives  helpful  sugges- 
tions for  the  exercises,  addressed  primarily  to  the  teacher,  and  indicating 
clearly  the  most  effective  pedagogical  methods. 


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A  New  Astronomy 

BY 

DAVID  P.  TODD,  M.A.,  Ph.D. 

Professor  of  Astronomy  and  Director  of  the  Observatory,  Amherst  College. 


Cloth,  i2mo,  480  pages.     Illustrated     -      -     Price,  $1.30 


This  book  is  designed  for  classes  pursuing  the  study  in 
High  Schools,  Academies,  and  Colleges.  The  author's 
long  experience  as  a  director  in  astronomical  observatories 
and  in  teaching  the  subject  has  given  him  unusual  qualifi- 
cations and  advantages  for  preparing  an  ideal  text-book. 

The  noteworthy  feature  which  distinguishes  this  from 
other  text-books  on  Astronomy  is  the  practical  way  in 
which  the  subjects  treated  are  enforced  by  laboratory 
experiments  and  methods.  In  this  the  author  follows  the 
principle  that  Astronomy  is  preeminently  a  science  of 
observation  and  should  be  so  taught. 

By  placing  more  importance  on  the  physical  than  on 
the  mathematical  facts  of  Astronomy  the  author  has  made 
ev6ry  page  of  the  book  deeply  interesting  to  the  student 
and  the  general  reader.  The  treatment  of  the  planets  and 
other  heavenly  bodies  and  of  the  law  of  universal  gravita- 
tion is  unusually  full,  clear,  and  illuminative.  The  mar- 
velous discoveries  of  Astronomy  in  recent  years,  and  the 
latest  advances  in  methods  of  teaching  the  science,  are 
all  represented. 

The  illustrations  are  an  important  feature  of  the  book. 
Many  of  them  are  so  ingeniously  devised  that  they  explain 
at  a  glance  what  pages  of  mere  description  could  not  make 
clear.  

Copies  of  Todd's  New  Astronomy  will  be  sent.,  prepaid.,  to  any  address 
on  receipt  of  the  price  by  the  Publishers  : 

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